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Path: gt.sage
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#GRAPH UTILITIES
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def check_independence_extension(g,S):
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"""
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Returns True if the set S extends to a maximum independent set of the graph g.
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sage: check_independence_extension(graphs.CycleGraph(6), Set([0,2]))
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True
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sage: check_independence_extension(graphs.CycleGraph(6), Set([0,3]))
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False
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"""
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V = g.vertices()
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alpha = g.independent_set(value_only=True)
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#print alpha
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if not S.issubset(Set(V)) or not g.is_independent_set(S):
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return False
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N = neighbors_set(g,S)
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X = [v for v in V if v not in S and v not in N]
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h = g.subgraph(X)
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alpha_h = h.independent_set(value_only=True)
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#print alpha_h, len(S)
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return (alpha == alpha_h + len(S))
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def find_alpha_critical_graphs(order, save = False):
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"""
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Returns a list of the graph6 string of each of the alpha critical graphs of
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the given order. A graph g is alpha critical if alpha(g-e) > alpha(g) for
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every edge e in g. This looks at every graph of the given order, so this
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will be slow for any order larger than 8.
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If save = True (default False), then list will be saved to a file
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named "alpha_critical_name_list_{order}".
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There is a unique alpha critical graph on 3 and 4 vertices::
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sage: find_alpha_critical_graphs(3)
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['Bw']
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sage: find_alpha_critical_graphs(4)
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['C~']
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There are two alpha critical graphs on 5 vertices::
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sage: find_alpha_critical_graphs(5)
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['Dhc', 'D~{']
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There are two alpha critical graphs on 6 vertices::
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sage: find_alpha_critical_graphs(6)
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['E|OW', 'E~~w']
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"""
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graphgen = graphs(order)
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alpha_critical_name_list = []
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for g in graphgen:
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if g.is_connected():
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if is_alpha_critical(g):
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alpha_critical_name_list.append(g.graph6_string())
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s = "alpha_critical_name_list_{}".format(order)
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if save:
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save(alpha_critical_name_list, s)
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return alpha_critical_name_list
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def is_degree_sequence(L):
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"""
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Returns True if the list L is the degree sequence of some graph.
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Since a graph always contains at least two vertices of the same degree, a
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list containing no duplicates cannot be a degree sequence::
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sage: is_degree_sequence([i for i in range(8)])
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False
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A cycle has all degrees equal to two and exists for any order larger than
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3, so a list of twos of length at least 3 is a degree sequence::
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sage: is_degree_sequence([2]*10)
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True
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"""
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try:
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graphs.DegreeSequence(L)
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except:
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return False
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return True
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#ALPHA APPROXIMATIONS
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def find_lower_bound_sets(g, i):
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"""
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Returns a list of independent sets of size i unioned with their neighborhoods.
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Since this checks all subsets of size i, this is a potentially slow method!
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sage: l = find_lower_bound_sets(graphs.CycleGraph(6),2)
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sage: l
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[{0, 1, 2, 3, 5},
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{0, 1, 2, 3, 4, 5},
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{0, 1, 3, 4, 5},
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{0, 1, 2, 3, 4},
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{0, 1, 2, 4, 5},
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{1, 2, 3, 4, 5},
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{0, 2, 3, 4, 5}]
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sage: type(l[0])
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<class 'sage.sets.set.Set_object_enumerated_with_category'>
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"""
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V = g.vertices()
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lowersets = []
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for S in Subsets(Set(V),i):
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if g.is_independent_set(S):
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T = Set(closed_neighborhood(g,list(S)))
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if T not in Set(lowersets):
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lowersets.append(T)
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return lowersets
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def alpha_lower_approximation(g, i):
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LP = MixedIntegerLinearProgram(maximization=False)
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x = LP.new_variable(nonnegative=True)
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# We define the objective
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LP.set_objective(sum([x[v] for v in g]))
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# For any subset, we define a constraint
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for j in range(1,i+1):
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for S in find_lower_bound_sets(g, j):
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#print S, S.cardinality()
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LP.add_constraint(sum([x[k] for k in S]), min = j)
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LP.solve()
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#return LP
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x_sol = LP.get_values(x)
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print x_sol
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return sum(x_sol.values())
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#input = graph g
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#output = bipartite graph with twice as many nodes and edges
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#new nodes are labeled n to 2n-1
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#assumes nodes in g are labeled [0..n-1]
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#same as cartesian product with k2, but output labeling is guarnateed to be integers
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def make_bidouble_graph(g):
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n = g.order()
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gdub = Graph(2*n)
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#print "gdub order = {}".format(gdub.order())
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for (i,j) in g.edges(labels = False):
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#print (i,j)
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gdub.add_edge(i,j+n)
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gdub.add_edge(j,i+n)
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return gdub
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def neighbors_set(g,S):
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N = set()
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for v in S:
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for n in g.neighbors(v):
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N.add(n)
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return list(N)
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def closed_neighborhood(g, verts):
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if isinstance(verts, list):
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neighborhood = []
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for v in verts:
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neighborhood += [v] + g.neighbors(v)
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return list(set(neighborhood))
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else:
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return [verts] + g.neighbors(verts)
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def is_alpha_critical(g):
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#if not g.is_connected():
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#return False
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alpha = g.independent_set(value_only=True)
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for e in g.edges():
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gc = copy(g)
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gc.delete_edge(e)
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alpha_prime = gc.independent_set(value_only=True)
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if alpha_prime <= alpha:
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return False
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return True
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#HEURISTIC ALGORITHMS
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#takes vertex of max degree, deletes so long as degree > 0, returns remaining ind set
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def MAXINE_independence_heuristic(g):
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V = g.vertices()
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h = g.subgraph(V)
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delta = max(h.degree())
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while delta > 0:
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#print "V is {}".format(V)
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#print "h vertices = {}, h.degree = {}".format(h.vertices(),h.degree())
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max_degree_vertex = V[h.degree().index(delta)]
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#print "max_degree_vertex = {}".format(max_degree_vertex)
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#print "h.neighbors(max_degree_vertex) = {}".format(h.neighbors(max_degree_vertex))
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V.remove(max_degree_vertex)
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h = g.subgraph(V)
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delta = max(h.degree())
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print "delta = {}".format(delta)
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return len(V)
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#takes vertex of min degree, adds it to max ind set until no vertices left
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def MIN_independence_heuristic(g):
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V = g.vertices()
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I = []
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while V != []:
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#print "V is {}".format(V)
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h = g.subgraph(V)
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#print "h vertices = {}, h.degree = {}".format(h.vertices(),h.degree())
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delta = min(h.degree())
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#print "delta = {}".format(delta)
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min_degree_vertex = V[h.degree().index(delta)]
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#print "min_degree_vertex = {}".format(min_degree_vertex)
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I.append(min_degree_vertex)
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V = [v for v in V if v not in closed_neighborhood(h, min_degree_vertex)]
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#break
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return len(I)
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"""
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Returns true if the given graph exists in the given list.
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It also prints out all graphs in the list that are isomorphic so that duplicates may also be found here.
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"""
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def does_graph_exist(g, L):
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success = False
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for gL in L:
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if g.is_isomorphic(gL):
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print gL.name()
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success = True
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return success
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"""
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Returns a list of all pairs of isomorphic graphs in the given list.
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"""
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import itertools
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def find_isomorphic_pairs(l):
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pairs = []
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L = itertools.combinations(l, r = 2)
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for pair in L:
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if pair[0].is_isomorphic(pair[1]):
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pairs.append(pair)
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return pairs
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def find_all_max_ind_sets(g):
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"""
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Finds all the maximum independent sets and stores them in a list
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"""
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final_list = []
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V = Set(g.vertices())
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alpha = independence_number(g)
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for s in V.subsets(alpha):
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if g.is_independent_set(s):
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final_list.append(s)
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return final_list
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def add_to_lists(graph, *L):
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"""
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Adds the specified graph to the arbitrary number of lists given as the second through last argument
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Use this function to build the lists of graphs
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"""
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for list in L:
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list.append(graph)
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def MIR(n):
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if n < 2:
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raise RuntimeError("MIR is defined for n >= 2")
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if n % 2 == 0:
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g = graphs.PathGraph(2)
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else:
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g = graphs.PathGraph(3)
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while g.order() < n:
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new_v = g.add_vertex()
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for v in g.vertices():
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if v != new_v:
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g.add_edge(v, new_v)
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g.add_edge(new_v, g.add_vertex())
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return g
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def Ciliate(q, r):
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if q < 1:
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raise RuntimeError("q must be greater than or equal to 1")
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if r < q:
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raise RuntimeError("r must be greater than or equal to q")
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if q == 1:
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return graphs.PathGraph(2*r)
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if q == r:
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return graphs.CycleGraph(2*q)
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g = graphs.CycleGraph(2*q)
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for v in g.vertices():
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g.add_path([v]+[g.add_vertex() for _ in range(r-q)])
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return g
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def Antihole(n):
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if n < 5:
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raise RuntimeError("antihole is defined for n > 5")
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return graphs.CycleGraph(n).complement()
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def Caro_Roditty(n):
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"""
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p.171
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Caro, Y., and Y. Roditty. "On the vertex-independence number and star decomposition of graphs." Ars Combinatoria 20 (1985): 167-180.
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"""
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g = graphs.CycleGraph(4)
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iters = 1
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while iters < n:
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len_v = len(g.vertices())
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g.add_cycle(range(len_v, len_v + 4))
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last_cycle = g.vertices()[-4:]
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for v in last_cycle:
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g.add_edge(v, v-4)
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iters += 1
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return g
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def find_all_triangles(g):
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E = g.edges()
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A = g.adjacency_matrix()
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pos = {v:p for (p,v) in enumerate(g.vertices())}
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triangles = []
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for e in E:
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v,w = (e[0], e[1]) if pos[e[0]] < pos[e[1]] else (e[1], e[0])
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S = [u for u in g.vertices() if g.has_edge(u,v) and g.has_edge(u,v) and pos[u] > pos[w]]
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for u in S:
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s = Set([u,v,w])
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triangles.append(s)
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return triangles
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# the triangles of a graph g are the vertices of the returned auxilliary graph aux_g
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# with edges in aux_g between a pair of vertices in aux_g if the corresponding triangles share a vertex of g
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def form_triangles_graph(g):
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vertices = find_all_triangles(g)
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edges = []
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for i in range(len(vertices)-1):
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for j in range(i+1, len(vertices)):
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if not((vertices[i].intersection(vertices[j])).is_empty()):
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edges.append((vertices[i],vertices[j]))
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return Graph([vertices,edges])
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def max_bipartite_set(g,s,c):
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#print "s is {}".format(s)
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#print "c is {}".format(c)
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if len(c) == 0:
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return s
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v = c[0]
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#print "v is {}".format(v)
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SCopy = copy(s)
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SCopy.append(v)
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Gprime = g.subgraph(SCopy)
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CCopy = copy(c)
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CCopy.remove(v) #CCopy is C with v removed
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if not(Gprime.is_bipartite()):
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#print "{} is not bipartite".format(SCopy)
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return max_bipartite_set(g, s, CCopy)
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temp1 = max_bipartite_set(g, SCopy, CCopy)
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temp2 = max_bipartite_set(g, s, CCopy)
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if len(temp1) > len(temp2):
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return temp1
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else:
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return temp2
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# output = closure of input graph
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# Useful: a graph is hamiltonian iff its closure is hamiltonian
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def closure(graph):
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"""
372
Test cases:
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sage: closure(graphs.CycleGraph(4)).is_isomorphic(graphs.CompleteGraph(4))
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True
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sage: closure(graphs.CycleGraph(5)).is_isomorphic(graphs.CycleGraph(5))
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True
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"""
378
from itertools import combinations
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g = graph.copy()
380
while(True):
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flag = False
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deg = g.degree()
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for (v,w) in combinations(g.vertices(), 2):
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if (not g.has_edge(v,w)) and deg[v] + deg[w] >= g.order():
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g.add_edge(v,w)
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flag = True
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if flag == False:
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break
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return g
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def is_simplical_vertex(g, v):
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"""
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Vertex v is a simplical vertex in g if the induced neighborhood of v is a clique
394
"""
395
neighbors = g.neighbors(v)
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induced_neighborhood = g.subgraph(neighbors)
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return induced_neighborhood.is_clique()
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399
# Defined by Sergey Norin at SIAM DM 2018
400
def is_homogenous_set(g, s):
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"""
402
Set of vertices s is homogenous if s induces a clique or s is an independent set.
403
"""
404
induced_g = g.subgraph(s)
405
return g.is_independent_set(s) or induced_g.is_clique()
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407
def generalized_degree(g,S):
408
"""
409
The cardinality of the union of the neighborhoods of each v in S.
410
"""
411
neighborhood_union = set(w for v in S for w in g.neighbors(v))
412
return len(neighborhood_union)
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414
def common_neighbors_of_set(g, s):
415
"""
416
Returns the vertices in g adjacent to every vertex in s
417
"""
418
if not s:
419
return []
420
comm_neigh = set(g.neighbors(s[0]))
421
for v in s[1:]:
422
comm_neigh = comm_neigh.intersection(set(g.neighbors(v)))
423
return list(comm_neigh)
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425
def common_neighbors(g, v, w):
426
"""
427
returns the Set of common neighbors of v and w in graph g
428
sage: common_neighbors(p4, 0, 3)
429
{}
430
sage: common_neighbors(p4, 0, 2)
431
{1}
432
"""
433
Nv = Set(g.neighbors(v))
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Nw = Set(g.neighbors(w))
435
return Nv.intersection(Nw)
436
437
def extremal_triangle_free_extension(g):
438
"""
439
Returns graph with edges added until no more possible without creating triangles.
440
If input is not triangle-free, raises RuntimeError.
441
442
This function is not deterministic; the output may vary among any of possibly many extremal triangle-free extensions.
443
The output is also not necessarily the maximal triangle-free extension.
444
"""
445
if not g.is_triangle_free():
446
raise RuntimeError("Graph is not triangle-free")
447
448
g2 = g.copy()
449
from itertools import combinations
450
for (v,w) in combinations(sample(g2.vertices(), k = g2.order()), 2): # Sample so output not deterministic
451
if not g2.has_edge(v, w) and all(u not in g2.neighbors(v) for u in g2.neighbors(w)):
452
g2.add_edge(v, w)
453
return g2
454
455
def pyramid_encapsulation(g):
456
"""
457
Returns the pyramid encapsulation of graph g.
458
459
Let a pyramid be a triangle with each edge bisected, and the midpoints
460
joined to form an inner triangle on vertices 0,1,2
461
For any graph g, make all its vertices adjacent to 0,1,2.
462
463
The pyramid is a pebbling Class1 graph (pebbling number is order + 1).
464
The pyramid encapuslation always yields a Class1 graph.
465
"""
466
pyramid = graphs.CompleteGraph(3)
467
pyramid.add_vertices([3, 4, 5])
468
pyramid.add_edges([[3,1], [3,0], [4,1], [4,2], [5,0], [5,2]])
469
470
pe = pyramid.disjoint_union(g)
471
for v in [0, 1, 2]:
472
for w in g.vertices():
473
pe.add_edge((0, v), (1,w))
474
return pe
475
476
def cycle_lengths(g):
477
"""
478
Returns set of all cycle lengths in g - without repetition
479
480
If g is acyclic, returns an empty list.
481
Performs depth-first search of all possible cycles.
482
"""
483
lengths = set()
484
for init_vertex in g.vertices():
485
path_stack = [[init_vertex]]
486
while path_stack:
487
path = path_stack.pop()
488
for neighbor in g.neighbors(path[-1]):
489
if neighbor not in path:
490
path_stack.append(path + [neighbor])
491
elif neighbor == path[0] and len(path) > 2:
492
lengths.add(len(path))
493
return lengths
494
495
def max_induced_tree(g):
496
"""
497
Returns *a* maximum-size tree which is an induced subgraph of g
498
499
Raises ValueError if g is not connected, since some invariant theorems assume connected.
500
"""
501
if not g.is_connected():
502
raise ValueError("Input graph is not connected")
503
504
from itertools import combinations
505
for j in xrange(g.order()):
506
for subset in combinations(sample(g.vertices(), k = g.order()), j): # randomize so avg.-case time, not worst-case
507
sub_g = g.copy()
508
sub_g.delete_vertices(subset)
509
if sub_g.is_tree():
510
return sub_g
511
512
def max_induced_forest(g):
513
"""
514
Returns *a* maximum-size induced subgraph of g which is a forest
515
516
Accepts both connected and disconnected graphs as input.
517
"""
518
from itertools import combinations
519
for j in xrange(g.order()):
520
for subset in combinations(sample(g.vertices(), k = g.order()), j): # randomize so avg.-case time, not worst-case
521
sub_g = g.copy()
522
sub_g.delete_vertices(subset)
523
if sub_g.is_forest():
524
return sub_g
525
526
def is_matching(s):
527
"""
528
True if set of edges s is a matching, i.e. no edges share a common vertex
529
530
Ignores edges labels; only compares indices 0 and 1 in edge tuples.
531
"""
532
vertex_list = [v for e in s for v in e[:2]] # Ignore any labels
533
if len(vertex_list) != len(set(vertex_list)):
534
return False
535
else:
536
return True
537
538
def mobius_ladder(k):
539
"""
540
A mobius ladder with parameter k is a cubic graph on 2k vertices which can
541
be constructed by taking a cycle on 2k vertices and connecting opposite
542
vertices.
543
544
sage: ml10 = mobius_ladder(10)
545
sage: ml10
546
mobius_ladder_10: Graph on 20 vertices
547
sage: ml10.order()
548
20
549
sage: ml10.degree()
550
[3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3]
551
sage: ml10.is_apex()
552
True
553
sage: ml10.is_vertex_transitive()
554
True
555
"""
556
g = graphs.CycleGraph(2*k)
557
for i in range(k):
558
g.add_edge(i, i+k)
559
g.name(new = "mobius_ladder_{}".format(k))
560
return g
561
562
def benoit_boyd_graphs(a, b, c):
563
"""
564
Two triangles pointed at eachother, with opposite vertices connected by paths of a,b,c respective edges. Triangles weighted 0.5, paths 1.0.
565
566
Pg. 927 of Geneviève Benoit and Sylvia Boyd, Finding the Exact Integrality Gap for Small Traveling Salesman Problems.
567
Mathematics of Operations Research, 33(4): 921--931, 2008.
568
"""
569
g = Graph(0, weighted = True)
570
for i in xrange(0, a):
571
g.add_edge(i, i + 1, 1)
572
for i in xrange(a + 1, a + b + 1):
573
g.add_edge(i, i + 1, 1)
574
for i in xrange(a + b + 2, a + b + c + 2):
575
g.add_edge(i, i + 1, 1)
576
g.add_edges([(0, a + 1, 0.5), (a + 1, a + b + 2, 0.5), (0, a + b + 2, 0.5)])
577
g.add_edges([(a, a + b + 1, 0.5), (a + b + 1, a + b + c + 2, 0.5), (a, a + b + c + 2, 0.5)])
578
return g
579
580
def benoit_boyd_graphs_2(a, b, c):
581
"""
582
Two triangles pointed at eachother, with opposite vertices connected by paths of a,b,c respective edges. Weights more complicated.
583
584
Paths each weighted 1/a, 1/b, 1/c. The triangles are weighted with the sum of the paths they join, e.g. 1/a+1/b or 1/b+1/c.
585
586
Pg. 928 of Geneviève Benoit and Sylvia Boyd, Finding the Exact Integrality Gap for Small Traveling Salesman Problems.
587
Mathematics of Operations Research, 33(4): 921--931, 2008.
588
"""
589
g = Graph(0, weighted = True)
590
for i in xrange(0, a):
591
g.add_edge(i, i + 1, 1/a)
592
for i in xrange(a + 1, a + b + 1):
593
g.add_edge(i, i + 1, 1/b)
594
for i in xrange(a + b + 2, a + b + c + 2):
595
g.add_edge(i, i + 1, 1/c)
596
g.add_edges([(0, a + 1, 1/a + 1/b), (a + 1, a + b + 2, 1/b + 1/c), (0, a + b + 2, 1/a + 1/c)])
597
g.add_edges([(a, a + b + 1, 1/a + 1/b), (a + b + 1, a + b + c + 2, 1/b + 1/c), (a, a + b + c + 2, 1/a + 1/c)])
598
return g
599
600
def bipartite_double_cover(g):
601
"""
602
Returns the bipatite double cover of a graph ``g``.
603
604
From :wikipedia:`Bipartite double cover`:
605
The bipartite double cover of ``g`` may also be known as the
606
Kronecker double cover, canonical double cover or the bipartite double of G.
607
For every vertex `v_i` of ``g``, there are two vertices `u_i` and `w_i`.
608
Two vertices `u_i` and `w_j` are connected by an edge in the double cover if
609
and only if `v_i` and `v_j` are connected by an edge in ``g``.
610
611
EXAMPLES:
612
613
sage: bipartite_double_cover(graphs.PetersenGraph()).is_isomorphic(graphs.DesarguesGraph())
614
True
615
616
sage: bipartite_double_cover(graphs.CycleGraph(4)).is_isomorphic(graphs.CycleGraph(4).disjoint_union(graphs.CycleGraph(4)))
617
True
618
"""
619
return g.tensor_product(graphs.CompleteGraph(2))
620
621
#TESTING
622
623
#check for invariant relation that separtates G from class defined by property
624
def find_separating_invariant_relation(g, objects, property, invariants):
625
L = [x for x in objects if (property)(x)]
626
for inv1 in invariants:
627
for inv2 in invariants:
628
if inv1(g) > inv2(g) and all(inv1(x) <= inv2(x) for x in L):
629
return inv1.__name__, inv2.__name__
630
print "no separating invariants"
631
632
633
634
#finds "difficult" graphs for necessary conditions, finds graphs which don't have property but which have all necessary conditions
635
def test_properties_upper_bound_theory(objects, property, theory):
636
for g in objects:
637
if not property(g) and all(f(g) for f in theory):
638
print g.name()
639
640
#finds "difficult" graphs for sufficient conditions, finds graphs which dont have any sufficient but do have property
641
def test_properties_lower_bound_theory(objects, property, theory):
642
for g in objects:
643
if property(g) and not any(f(g) for f in theory):
644
print g.name()
645
646
def find_coextensive_properties(objects, properties):
647
for p1 in properties:
648
for p2 in properties:
649
if p1 != p2 and all(p1(g) == p2(g) for g in objects):
650
print p1.__name__, p2.__name__
651
print "DONE!"
652
653
print("loaded utilities")
654
655
#############################################################################
656
# End of utilities section #
657
#############################################################################
658
#GRAPH INVARIANTS
659
all_invariants = []
660
661
efficient_invariants = []
662
intractable_invariants = []
663
theorem_invariants = []
664
broken_invariants = []
665
666
"""
667
Last version of graphs packaged checked: Sage 8.2
668
sage: sage.misc.banner.version_dict()['major'] < 8 or (sage.misc.banner.version_dict()['major'] == 8 and sage.misc.banner.version_dict()['minor'] <= 2)
669
True
670
"""
671
sage_efficient_invariants = [Graph.number_of_loops, Graph.density, Graph.order, Graph.size, Graph.average_degree,
672
Graph.triangles_count, Graph.szeged_index, Graph.radius, Graph.diameter, Graph.girth, Graph.wiener_index,
673
Graph.average_distance, Graph.connected_components_number, Graph.maximum_average_degree, Graph.lovasz_theta,
674
Graph.spanning_trees_count, Graph.odd_girth, Graph.clustering_average, Graph.cluster_transitivity]
675
676
sage_intractable_invariants = [Graph.chromatic_number, Graph.chromatic_index, Graph.treewidth,
677
Graph.clique_number, Graph.pathwidth, Graph.fractional_chromatic_index, Graph.edge_connectivity,
678
Graph.vertex_connectivity, Graph.genus, Graph.crossing_number]
679
680
for i in sage_efficient_invariants:
681
add_to_lists(i, efficient_invariants, all_invariants)
682
for i in sage_intractable_invariants:
683
add_to_lists(i, intractable_invariants, all_invariants)
684
685
def distinct_degrees(g):
686
"""
687
returns the number of distinct degrees of a graph
688
sage: distinct_degrees(p4)
689
2
690
sage: distinct_degrees(k4)
691
1
692
"""
693
return len(set(g.degree()))
694
add_to_lists(distinct_degrees, efficient_invariants, all_invariants)
695
696
def max_common_neighbors(g):
697
"""
698
Returns the maximum number of common neighbors of any pair of distinct vertices in g.
699
700
sage: max_common_neighbors(p4)
701
1
702
sage: max_common_neighbors(k4)
703
2
704
"""
705
max = 0
706
V = g.vertices()
707
n = g.order()
708
for i in range(n):
709
for j in range(n):
710
if i < j:
711
temp = len(common_neighbors(g, V[i], V[j]))
712
if temp > max:
713
max = temp
714
return max
715
add_to_lists(max_common_neighbors, efficient_invariants, all_invariants)
716
717
def min_common_neighbors(g):
718
"""
719
Returns the minimum number of common neighbors of any pair of distinct vertices in g,
720
which is necessarily 0 for disconnected graphs.
721
722
sage: min_common_neighbors(p4)
723
0
724
sage: min_common_neighbors(k4)
725
2
726
"""
727
n = g.order()
728
min = n
729
V = g.vertices()
730
for i in range(n):
731
for j in range(n):
732
if i < j:
733
temp = len(common_neighbors(g, V[i], V[j]))
734
#if temp == 0:
735
#print "i={}, j={}".format(i,j)
736
if temp < min:
737
min = temp
738
return min
739
add_to_lists(min_common_neighbors, efficient_invariants, all_invariants)
740
741
def mean_common_neighbors(g):
742
"""
743
Returns the average number of common neighbors of any pair of distinct vertices in g.
744
sage: mean_common_neighbors(p4)
745
1/3
746
sage: mean_common_neighbors(k4)
747
2
748
"""
749
V = g.vertices()
750
n = g.order()
751
sum = 0
752
for i in range(n):
753
for j in range(n):
754
if i < j:
755
sum += len(common_neighbors(g, V[i], V[j]))
756
return 2*sum/(n*(n-1))
757
add_to_lists(mean_common_neighbors, efficient_invariants, all_invariants)
758
759
def min_degree(g):
760
"""
761
Returns the minimum of all degrees of the graph g.
762
763
sage: min_degree(graphs.CompleteGraph(5))
764
4
765
sage: min_degree(graphs.CycleGraph(5))
766
2
767
sage: min_degree(graphs.StarGraph(5))
768
1
769
sage: min_degree(graphs.CompleteBipartiteGraph(3,5))
770
3
771
"""
772
return min(g.degree())
773
add_to_lists(min_degree, efficient_invariants, all_invariants)
774
775
def max_degree(g):
776
"""
777
Returns the maximum of all degrees of the graph g.
778
779
sage: max_degree(graphs.CompleteGraph(5))
780
4
781
sage: max_degree(graphs.CycleGraph(5))
782
2
783
sage: max_degree(graphs.StarGraph(5))
784
5
785
sage: max_degree(graphs.CompleteBipartiteGraph(3,5))
786
5
787
"""
788
return max(g.degree())
789
add_to_lists(max_degree, efficient_invariants, all_invariants)
790
791
def median_degree(g):
792
"""
793
Return the median of the list of vertex degrees.
794
795
sage: median_degree(p4)
796
3/2
797
sage: median_degree(p3)
798
1
799
"""
800
return median(g.degree())
801
add_to_lists(median_degree, efficient_invariants, all_invariants)
802
803
def inverse_degree(g):
804
"""
805
Return the sum of the reciprocals of the non-zero degrees.
806
807
Return 0 if the graph has no edges.
808
809
sage: inverse_degree(p4)
810
3
811
sage: inverse_degree(graphs.CompleteGraph(1))
812
0
813
"""
814
if g.size() == 0:
815
return 0
816
return sum([(1.0/d) for d in g.degree() if d!= 0])
817
add_to_lists(inverse_degree, efficient_invariants, all_invariants)
818
819
def eulerian_faces(g):
820
"""
821
Returns 2 - order + size, which is the number of faces if the graph is planar,
822
a consequence of Euler's Formula.
823
824
sage: eulerian_faces(graphs.CycleGraph(5))
825
2
826
sage: eulerian_faces(graphs.DodecahedralGraph())
827
12
828
"""
829
n = g.order()
830
m = g.size()
831
return 2 - n + m
832
add_to_lists(eulerian_faces, efficient_invariants, all_invariants)
833
834
def barrus_q(g):
835
"""
836
If the degrees sequence is in non-increasing order, with index starting at 1,
837
barrus_q = max(k:d_k >= k)
838
839
Defined by M. Barrus in "Havel-Hakimi Residues of Unigraphs", 2012
840
841
sage: barrus_q(graphs.CompleteGraph(5))
842
4
843
sage: barrus_q(graphs.StarGraph(3))
844
1
845
"""
846
Degrees = g.degree()
847
Degrees.sort()
848
Degrees.reverse()
849
return max(k for k in range(g.order()) if Degrees[k] >= (k+1)) + 1
850
add_to_lists(barrus_q, efficient_invariants, all_invariants)
851
852
def barrus_bound(g):
853
"""
854
Returns n - barrus q
855
856
Defined in: Barrus, Michael D. "Havel–Hakimi residues of unigraphs." Information Processing Letters 112.1 (2012): 44-48.
857
858
sage: barrus_bound(k4)
859
1
860
sage: barrus_bound(graphs.OctahedralGraph())
861
2
862
"""
863
return g.order() - barrus_q(g)
864
add_to_lists(barrus_bound, efficient_invariants, all_invariants)
865
866
def matching_number(g):
867
"""
868
Returns the matching number of the graph g, i.e., the size of a maximum
869
matching.
870
871
A matching is a set of independent edges.
872
873
See: https://en.wikipedia.org/wiki/Matching_(graph_theory)
874
875
sage: matching_number(graphs.CompleteGraph(5))
876
2
877
sage: matching_number(graphs.CycleGraph(5))
878
2
879
sage: matching_number(graphs.StarGraph(5))
880
1
881
sage: matching_number(graphs.CompleteBipartiteGraph(3,5))
882
3
883
"""
884
return int(g.matching(value_only=True, use_edge_labels=False))
885
add_to_lists(matching_number, efficient_invariants, all_invariants)
886
887
def residue(g):
888
"""
889
If the Havel-Hakimi process is iterated until a sequence of 0s is returned,
890
residue is defined to be the number of zeros of this sequence.
891
892
See: Favaron, Odile, Maryvonne Mahéo, and J‐F. Saclé. "On the residue of a graph." Journal of Graph Theory 15.1 (1991): 39-64.
893
894
sage: residue(k4)
895
1
896
sage: residue(p4)
897
2
898
"""
899
seq = g.degree_sequence()
900
901
while seq[0] > 0:
902
d = seq.pop(0)
903
seq[:d] = [k-1 for k in seq[:d]]
904
seq.sort(reverse=True)
905
906
return len(seq)
907
add_to_lists(residue, efficient_invariants, all_invariants)
908
909
def annihilation_number(g):
910
"""
911
Given the degree sequence in non-degreasing order, with indices starting at 1, the annihilation number is the largest index k so the sum of the first k degrees is no more than the sum of the remaining degrees
912
913
See: Larson, Craig E., and Ryan Pepper. "Graphs with equal independence and annihilation numbers." the electronic journal of combinatorics 18.1 (2011): 180.
914
915
sage: annihilation_number(c4)
916
2
917
sage: annihilation_number(p5)
918
3
919
"""
920
seq = sorted(g.degree())
921
922
a = 0
923
while sum(seq[:a+1]) <= sum(seq[a+1:]):
924
a += 1
925
926
return a
927
add_to_lists(annihilation_number, efficient_invariants, all_invariants)
928
929
def fractional_alpha(g):
930
"""
931
This is the optimal solution of the linear programming relaxation of the integer programming formulation of independence number (alpha).
932
933
See: Nemhauser, George L., and Leslie Earl Trotter. "Vertex packings: structural properties and algorithms." Mathematical Programming 8.1 (1975): 232-248.
934
935
sage: fractional_alpha(k3)
936
1.5
937
sage: fractional_alpha(p5)
938
3.0
939
"""
940
if len(g.vertices()) == 0:
941
return 0
942
p = MixedIntegerLinearProgram(maximization=True)
943
x = p.new_variable(nonnegative=True)
944
p.set_objective(sum(x[v] for v in g.vertices()))
945
946
for v in g.vertices():
947
p.add_constraint(x[v], max=1)
948
949
for (u,v) in g.edge_iterator(labels=False):
950
p.add_constraint(x[u] + x[v], max=1)
951
952
return p.solve()
953
add_to_lists(fractional_alpha, efficient_invariants, all_invariants)
954
955
def fractional_covering(g):
956
"""
957
This is the optimal solution of the linear programming relaxation of the integer programming formulation of covering number.
958
959
For ILP formulation see: https://en.wikipedia.org/wiki/Vertex_cover
960
961
sage: fractional_covering(k3)
962
1.5
963
sage: fractional_covering(p5)
964
2.0
965
"""
966
if len(g.vertices()) == 0:
967
return 0
968
p = MixedIntegerLinearProgram(maximization=False)
969
x = p.new_variable(nonnegative=True)
970
p.set_objective(sum(x[v] for v in g.vertices()))
971
972
for v in g.vertices():
973
p.add_constraint(x[v], max=1)
974
975
for (u,v) in g.edge_iterator(labels=False):
976
p.add_constraint(x[u] + x[v], min=1)
977
978
return p.solve()
979
add_to_lists(fractional_covering, efficient_invariants, all_invariants)
980
981
def cvetkovic(g):
982
"""
983
This in the minimum of the number of nonnegative and nonpositive eigenvalues of the adjacency matrix.
984
985
Cvetkovic's theorem says that this number is an upper bound for the independence number of a graph.
986
987
See: Cvetković, Dragoš M., Michael Doob, and Horst Sachs. Spectra of graphs: theory and application. Vol. 87. Academic Pr, 1980.
988
989
sage: cvetkovic(p5)
990
3
991
sage: cvetkovic(graphs.PetersenGraph())
992
4
993
"""
994
eigenvalues = g.spectrum()
995
positive = 0
996
negative = 0
997
zero = 0
998
for e in eigenvalues:
999
if e > 0:
1000
positive += 1
1001
elif e < 0:
1002
negative += 1
1003
else:
1004
zero += 1
1005
1006
return zero + min([positive, negative])
1007
add_to_lists(cvetkovic, efficient_invariants, all_invariants)
1008
1009
def cycle_space_dimension(g):
1010
"""
1011
Returns the dimension of the cycle space (also called the circuit rank).
1012
1013
See: https://en.wikipedia.org/wiki/Cycle_space
1014
And: https://en.wikipedia.org/wiki/Circuit_rank
1015
1016
sage: cycle_space_dimension(k3)
1017
1
1018
sage: cycle_space_dimension(c4c4)
1019
2
1020
sage: cycle_space_dimension(glasses_5_5)
1021
2
1022
"""
1023
return g.size()-g.order()+g.connected_components_number()
1024
add_to_lists(cycle_space_dimension, efficient_invariants, all_invariants)
1025
1026
def card_center(g):
1027
return len(g.center())
1028
add_to_lists(card_center, efficient_invariants, all_invariants)
1029
1030
def card_periphery(g):
1031
return len(g.periphery())
1032
add_to_lists(card_periphery, efficient_invariants, all_invariants)
1033
1034
def max_eigenvalue(g):
1035
return max(g.adjacency_matrix().change_ring(RDF).eigenvalues())
1036
add_to_lists(max_eigenvalue, efficient_invariants, all_invariants)
1037
1038
def min_eigenvalue(g):
1039
return min(g.adjacency_matrix().change_ring(RDF).eigenvalues())
1040
add_to_lists(min_eigenvalue, efficient_invariants, all_invariants)
1041
1042
def resistance_distance_matrix(g):
1043
L = g.laplacian_matrix()
1044
n = g.order()
1045
J = ones_matrix(n,n)
1046
temp = L+(1.0/n)*J
1047
X = temp.inverse()
1048
R = (1.0)*ones_matrix(n,n)
1049
for i in range(n):
1050
for j in range(n):
1051
R[i,j] = X[i,i] + X[j,j] - 2*X[i,j]
1052
return R
1053
1054
def kirchhoff_index(g):
1055
R = resistance_distance_matrix(g)
1056
return .5*sum(sum(R))
1057
add_to_lists(kirchhoff_index, efficient_invariants, all_invariants)
1058
1059
def largest_singular_value(g):
1060
A = matrix(RDF,g.adjacency_matrix(sparse=False))
1061
svd = A.SVD()
1062
sigma = svd[1]
1063
return sigma[0,0]
1064
add_to_lists(largest_singular_value, efficient_invariants, all_invariants)
1065
1066
def card_max_cut(g):
1067
return g.max_cut(value_only=True)
1068
add_to_lists(card_max_cut, intractable_invariants, all_invariants)
1069
1070
def welsh_powell(g):
1071
"""
1072
for degrees d_1 >= ... >= d_n
1073
returns the maximum over all indices i of of the min(i,d_i + 1)
1074
1075
sage: welsh_powell(k5) = 4
1076
"""
1077
n= g.order()
1078
D = g.degree()
1079
D.sort(reverse=True)
1080
mx = 0
1081
for i in range(n):
1082
temp = min({i,D[i]})
1083
if temp > mx:
1084
mx = temp
1085
return mx + 1
1086
add_to_lists(welsh_powell, efficient_invariants, all_invariants)
1087
1088
#outputs upper bound from Brooks Theorem: returns Delta + 1 for complete and odd cycles
1089
def brooks(g):
1090
Delta = max(g.degree())
1091
delta = min(g.degree())
1092
n = g.order()
1093
if is_complete(g):
1094
return Delta + 1
1095
elif n%2 == 1 and g.is_connected() and Delta == 2 and delta == 2: #same as if all degrees are 2
1096
return Delta + 1
1097
else:
1098
return Delta
1099
add_to_lists(brooks, efficient_invariants, all_invariants)
1100
1101
#wilf's upper bound for chromatic number
1102
def wilf(g):
1103
return max_eigenvalue(g) + 1
1104
add_to_lists(wilf, efficient_invariants, all_invariants)
1105
1106
#a measure of irregularity
1107
def different_degrees(g):
1108
return len(set(g.degree()))
1109
add_to_lists(different_degrees, efficient_invariants, all_invariants)
1110
1111
def szekeres_wilf(g):
1112
"""
1113
Returns 1+ max of the minimum degrees for all subgraphs
1114
Its an upper bound for chromatic number
1115
1116
sage: szekeres_wilf(graphs.CompleteGraph(5))
1117
5
1118
"""
1119
#removes a vertex, if possible, of degree <= i
1120
def remove_vertex_of_degree(gc,i):
1121
Dc = gc.degree()
1122
V = gc.vertices()
1123
#print "Dc is %s, V is %s" %(Dc,V)
1124
mind = min(Dc)
1125
#print "mind is %s" %mind
1126
if mind <= i:
1127
1128
ind = Dc.index(mind)
1129
#print "ind is %s, vertex is %s" %(ind,V[ind])
1130
return gc.delete_vertex(V[ind])
1131
else:
1132
return gc
1133
D = g.degree()
1134
delta = min(D)
1135
Delta = max(D)
1136
for i in range(delta,Delta+1):
1137
gc = copy(g)
1138
value = g.order() + 1
1139
while gc.size() > 0 and gc.order() < value:
1140
#gc.show()
1141
value = gc.order()
1142
remove_vertex_of_degree(gc,i)
1143
if gc.size() == 0:
1144
return i + 1
1145
add_to_lists(szekeres_wilf, efficient_invariants, all_invariants)
1146
1147
def average_vertex_temperature(g):
1148
D = g.degree()
1149
n = g.order()
1150
return sum([D[i]/(n-D[i]+0.0) for i in range(n)])/n
1151
add_to_lists(average_vertex_temperature, efficient_invariants, all_invariants)
1152
1153
def sum_temperatures(g):
1154
D = g.degree()
1155
n = g.order()
1156
return sum([D[i]/(n-D[i]+0.0) for i in range(n)])
1157
add_to_lists(sum_temperatures, efficient_invariants, all_invariants)
1158
1159
def randic(g):
1160
D = g.degree()
1161
V = g.vertices()
1162
if min(D) == 0:
1163
return oo
1164
sum = 0
1165
for e in g.edges():
1166
v = e[0]
1167
i = V.index(v)
1168
w = e[1]
1169
j = V.index(w)
1170
sum += 1.0/(D[i]*D[j])**0.5
1171
return sum
1172
add_to_lists(randic, efficient_invariants, all_invariants)
1173
1174
#a very good lower bound for alpha
1175
def max_even_minus_even_horizontal(g):
1176
"""
1177
finds def max_even_minus_even_horizontal for each component and adds them up.
1178
"""
1179
mx_even=0
1180
Gcomps=g.connected_components_subgraphs()
1181
1182
while Gcomps != []:
1183
H=Gcomps.pop()
1184
temp=max_even_minus_even_horizontal_component(H)
1185
mx_even+=temp
1186
#print "temp = {}, mx_even = {}".format(temp,mx_even)
1187
1188
return mx_even
1189
add_to_lists(max_even_minus_even_horizontal, efficient_invariants, theorem_invariants, all_invariants)
1190
1191
def max_even_minus_even_horizontal_component(g):
1192
"""
1193
for each vertex v, find the number of vertices at even distance from v,
1194
and substract the number of edges induced by these vertices.
1195
this number is a lower bound for independence number.
1196
take the max. returns 0 if graph is not connected
1197
"""
1198
if g.is_connected()==False:
1199
return 0
1200
1201
distances = g.distance_all_pairs()
1202
mx=0
1203
for v in g.vertices():
1204
Even=[]
1205
for w in g.vertices():
1206
if distances[v][w]%2==0:
1207
Even.append(w)
1208
1209
#print len(Even), len(g.subgraph(Even).edges())
1210
l=len(Even)-len(g.subgraph(Even).edges())
1211
if l>mx:
1212
mx=l
1213
return mx
1214
1215
def laplacian_energy(g):
1216
L = g.laplacian_matrix().change_ring(RDF).eigenvalues()
1217
Ls = [1/lam**2 for lam in L if lam > 0]
1218
return 1 + sum(Ls)
1219
add_to_lists(laplacian_energy, efficient_invariants, all_invariants)
1220
1221
def laplacian_energy_like(g):
1222
"""
1223
Returns the sum of the square roots of the laplacian eigenvalues
1224
1225
Liu, Jianping, and Bolian Liu. "A Laplacian-energy-like invariant of a graph." MATCH-COMMUNICATIONS IN MATHEMATICAL AND IN COMPUTER CHEMISTRY 59.2 (2008): 355-372.
1226
"""
1227
return sum([sqrt(x) for x in g.spectrum(laplacian = True)])
1228
add_to_lists(laplacian_energy_like, efficient_invariants, all_invariants)
1229
1230
#sum of the positive eigenvalues of a graph
1231
def gutman_energy(g):
1232
L = g.adjacency_matrix().change_ring(RDF).eigenvalues()
1233
Ls = [lam for lam in L if lam > 0]
1234
return sum(Ls)
1235
add_to_lists(gutman_energy, efficient_invariants, all_invariants)
1236
1237
#the second smallest eigenvalue of the Laplacian matrix of a graph, also called the "algebraic connectivity" - the smallest should be 0
1238
def fiedler(g):
1239
L = g.laplacian_matrix().change_ring(RDF).eigenvalues()
1240
L.sort()
1241
return L[1]
1242
add_to_lists(fiedler, broken_invariants, all_invariants)
1243
1244
def degree_variance(g):
1245
mu = mean(g.degree())
1246
s = sum((x-mu)**2 for x in g.degree())
1247
return s/g.order()
1248
add_to_lists(degree_variance, efficient_invariants, all_invariants)
1249
1250
def graph_rank(g):
1251
return g.adjacency_matrix().rank()
1252
add_to_lists(graph_rank, efficient_invariants, all_invariants)
1253
1254
def card_positive_eigenvalues(g):
1255
return len([lam for lam in g.adjacency_matrix().eigenvalues() if lam > 0])
1256
add_to_lists(card_positive_eigenvalues, efficient_invariants, all_invariants)
1257
1258
def card_zero_eigenvalues(g):
1259
return len([lam for lam in g.adjacency_matrix().eigenvalues() if lam == 0])
1260
add_to_lists(card_zero_eigenvalues, efficient_invariants, all_invariants)
1261
1262
def card_negative_eigenvalues(g):
1263
return len([lam for lam in g.adjacency_matrix().eigenvalues() if lam < 0])
1264
add_to_lists(card_negative_eigenvalues, efficient_invariants, all_invariants)
1265
1266
def card_cut_vertices(g):
1267
return len((g.blocks_and_cut_vertices())[1])
1268
add_to_lists(card_cut_vertices, efficient_invariants, all_invariants)
1269
1270
def card_connectors(g):
1271
return g.order() - card_cut_vertices(g)
1272
add_to_lists(card_connectors, efficient_invariants, all_invariants)
1273
1274
#return number of leafs or pendants
1275
def card_pendants(g):
1276
return sum([x for x in g.degree() if x == 1])
1277
add_to_lists(card_pendants, efficient_invariants, all_invariants)
1278
1279
#returns number of bridges in graph
1280
def card_bridges(g):
1281
gs = g.strong_orientation()
1282
bridges = []
1283
for scc in gs.strongly_connected_components():
1284
bridges.extend(gs.edge_boundary(scc))
1285
return len(bridges)
1286
add_to_lists(card_bridges, efficient_invariants, all_invariants)
1287
1288
#upper bound for the domination number
1289
def alon_spencer(g):
1290
delta = min(g.degree())
1291
n = g.order()
1292
return n*((1+log(delta + 1.0)/(delta + 1)))
1293
add_to_lists(alon_spencer, efficient_invariants, all_invariants)
1294
1295
#lower bound for residue and, hence, independence number
1296
def caro_wei(g):
1297
return sum([1.0/(d + 1) for d in g.degree()])
1298
add_to_lists(caro_wei, efficient_invariants, all_invariants)
1299
1300
#equals 2*size, the 1st theorem of graph theory
1301
def degree_sum(g):
1302
return sum(g.degree())
1303
add_to_lists(degree_sum, efficient_invariants, all_invariants)
1304
1305
#smallest sum of degrees of non-adjacent degrees, invariant in ore condition for hamiltonicity
1306
#default for complete graph?
1307
def sigma_dist2(g):
1308
if g.size() == g.order()*(g.order()-1)/2:
1309
return g.order()
1310
Dist = g.distance_all_pairs()
1311
return min(g.degree(v) + g.degree(w) for v in g for w in g if Dist[v][w] > 1)
1312
add_to_lists(sigma_dist2, efficient_invariants, all_invariants)
1313
1314
#cardinality of the automorphism group of the graph
1315
def order_automorphism_group(g):
1316
return g.automorphism_group(return_group=False, order=True)
1317
add_to_lists(order_automorphism_group, efficient_invariants, all_invariants)
1318
1319
#in sufficient condition for graphs where vizing's independence theorem holds
1320
def brinkmann_steffen(g):
1321
E = g.edges()
1322
if len(E) == 0:
1323
return 0
1324
Dist = g.distance_all_pairs()
1325
return min(g.degree(v) + g.degree(w) for v in g for w in g if Dist[v][w] == 1)
1326
add_to_lists(brinkmann_steffen, efficient_invariants, all_invariants)
1327
1328
def alpha_critical_optimum(g, alpha_critical_names):
1329
1330
n = g.order()
1331
V = g.vertices()
1332
#g.show()
1333
1334
alpha_critical_graph_names = []
1335
1336
#get alpha_critical graphs with order <= n
1337
for name in alpha_critical_names:
1338
h = Graph(name)
1339
if h.order() <= n:
1340
alpha_critical_graph_names.append(h.graph6_string())
1341
1342
#print alpha_critical_graphs
1343
1344
LP = MixedIntegerLinearProgram(maximization=True)
1345
b = LP.new_variable(nonnegative=True)
1346
1347
# We define the objective
1348
LP.set_objective(sum([b[v] for v in g]))
1349
1350
# For any edge, we define a constraint
1351
for (u,v) in g.edges(labels=None):
1352
LP.add_constraint(b[u]+b[v],max=1)
1353
#LP.add_constraint(b[u]+b[v],min=1)
1354
1355
#search through all subsets of g with order >= 3
1356
#and look for *any* subgraph isomorphic to an alpha critical graph
1357
#for any match we define a constraint
1358
1359
i = 3
1360
while i <= n:
1361
SS = Subsets(Set(V),i)
1362
for S in SS:
1363
L = [g6 for g6 in alpha_critical_graph_names if Graph(g6).order() == i]
1364
#print L
1365
for g6 in L:
1366
h = Graph(g6)
1367
if g.subgraph(S).subgraph_search(h, induced=False):
1368
1369
#print S
1370
#add constraint
1371
alpha = independence_number(h)
1372
#print h.graph6_string(), alpha
1373
LP.add_constraint(sum([b[j] for j in S]), max = alpha, name = h.graph6_string())
1374
i = i + 1
1375
1376
#for c in LP.constraints():
1377
#print c
1378
1379
# The .solve() functions returns the objective value
1380
LP.solve()
1381
1382
#return LP
1383
1384
b_sol = LP.get_values(b)
1385
return b_sol, sum(b_sol.values())
1386
1387
1388
###several invariants and auxiliary functions related to the Independence Decomposition Theorem
1389
1390
#finds all vertices with weight 1 in some max weighted stable set with wieghts in {0,1,1/2}
1391
#had problem that LP solver has small numerical errors, fixed with kludgy if condition
1392
def find_stable_ones_vertices(g):
1393
F = []
1394
alpha_f = fractional_alpha(g)
1395
for v in g.vertices():
1396
gc = copy(g)
1397
gc.delete_vertices(closed_neighborhood(gc, v))
1398
alpha_f_prime = fractional_alpha(gc)
1399
if abs(alpha_f - alpha_f_prime - 1) < .01:
1400
F.append(v)
1401
return F
1402
1403
def find_max_critical_independent_set(g):
1404
S = find_stable_ones_vertices(g)
1405
H = g.subgraph(S)
1406
return H.independent_set()
1407
1408
def critical_independence_number(g):
1409
return len(find_max_critical_independent_set(g))
1410
add_to_lists(critical_independence_number, efficient_invariants, all_invariants)
1411
1412
def card_independence_irreducible_part(g):
1413
return len(find_independence_irreducible_part(g))
1414
add_to_lists(card_independence_irreducible_part, efficient_invariants, all_invariants)
1415
1416
def find_independence_irreducible_part(g):
1417
X = find_KE_part(g)
1418
SX = Set(X)
1419
Vertices = Set(g.vertices())
1420
return list(Vertices.difference(SX))
1421
1422
#returns KE part guaranteed by Independence Decomposition Theorem
1423
def find_KE_part(g):
1424
return closed_neighborhood(g, find_max_critical_independent_set(g))
1425
1426
def card_KE_part(g):
1427
return len(find_KE_part(g))
1428
add_to_lists(card_KE_part, efficient_invariants, all_invariants)
1429
1430
def find_independence_irreducible_subgraph(g):
1431
return g.subgraph(find_independence_irreducible_part(g))
1432
1433
def find_KE_subgraph(g):
1434
return g.subgraph(find_KE_part(g))
1435
1436
1437
#make invariant from property
1438
def make_invariant_from_property(property, name=None):
1439
"""
1440
This function takes a property as an argument and returns an invariant
1441
whose value is 1 if the object has the property, else 0
1442
Optionally a name for the new property can be provided as a second argument.
1443
"""
1444
def boolean_valued_invariant(g):
1445
if property(g):
1446
return 1
1447
else:
1448
return 0
1449
1450
if name is not None:
1451
boolean_valued_invariant.__name__ = name
1452
elif hasattr(property, '__name__'):
1453
boolean_valued_invariant.__name__ = '{}_value'.format(property.__name__)
1454
else:
1455
raise ValueError('Please provide a name for the new function')
1456
1457
return boolean_valued_invariant
1458
1459
# defined by R. Pepper in an unpublished paper on graph irregularity
1460
def geometric_length_of_degree_sequence(g):
1461
return sqrt(sum(d**2 for d in g.degree()))
1462
add_to_lists(geometric_length_of_degree_sequence, efficient_invariants, all_invariants)
1463
1464
# Two Stability Theta Bound
1465
# For graphs with alpha <= 2,
1466
# lovasz_theta <= 2^(2/3)*n^(1/3)
1467
# The Sandwich Theorem by Knuth p. 47
1468
def two_stability_theta_bound(g):
1469
return 2**(2/3)*g.order()**(1/3)
1470
add_to_lists(two_stability_theta_bound, efficient_invariants, all_invariants)
1471
1472
# Lovasz Theta over Root N
1473
# The Sandwich Theorem by Knuth p. 45
1474
def lovasz_theta_over_root_n(g):
1475
return g.lovasz_theta()/sqrt(g.order())
1476
add_to_lists(lovasz_theta_over_root_n, efficient_invariants, all_invariants)
1477
1478
# Theta * Theta-Complement
1479
# The Sandwich Theorem by Knuth, p. 27
1480
def theta_theta_complement(g):
1481
return g.lovasz_theta() * g.complement().lovasz_theta()
1482
add_to_lists(theta_theta_complement, efficient_invariants, all_invariants)
1483
1484
# Depth = Order - Residue
1485
# This is the number of steps it takes for Havel-Hakimi to terminate
1486
def depth(g):
1487
return g.order()-residue(g)
1488
add_to_lists(depth, efficient_invariants, all_invariants)
1489
1490
# Lovasz Theta of the complement of the given graph
1491
def lovasz_theta_complement(g):
1492
return g.complement().lovasz_theta()
1493
add_to_lists(lovasz_theta_complement, efficient_invariants, all_invariants)
1494
1495
# N over lovasz_theta_complement
1496
# This is a lower bound for lovasz theta
1497
# The Sandwich Theorem by Knuth, p. 27
1498
def n_over_lovasz_theta_complement(g):
1499
return g.order()/lovasz_theta_complement(g)
1500
add_to_lists(n_over_lovasz_theta_complement, efficient_invariants, all_invariants)
1501
1502
# The number of vertices at even distance from v and return the max over all vertices
1503
def max_even(g):
1504
from sage.graphs.distances_all_pairs import distances_all_pairs
1505
D = distances_all_pairs(g)
1506
evens_list = []
1507
for u in D:
1508
evens = 0
1509
for v in D[u]:
1510
if D[u][v] % 2 == 0:
1511
evens += 1
1512
evens_list.append(evens)
1513
return max(evens_list)
1514
add_to_lists(max_even, efficient_invariants, all_invariants)
1515
1516
# The number of vertices at even distance from v and return the min over all vertices
1517
def min_even(g):
1518
from sage.graphs.distances_all_pairs import distances_all_pairs
1519
D = distances_all_pairs(g)
1520
evens_list = []
1521
for u in D:
1522
evens = 0
1523
for v in D[u]:
1524
if D[u][v] % 2 == 0:
1525
evens += 1
1526
evens_list.append(evens)
1527
return min(evens_list)
1528
add_to_lists(min_even, efficient_invariants, all_invariants)
1529
1530
# The number of vertices at odd distance from v and return the max over all vertices
1531
def max_odd(g):
1532
from sage.graphs.distances_all_pairs import distances_all_pairs
1533
D = distances_all_pairs(g)
1534
odds_list = []
1535
for u in D:
1536
odds = 0
1537
for v in D[u]:
1538
if D[u][v] % 2 != 0:
1539
odds += 1
1540
odds_list.append(odds)
1541
return max(odds_list)
1542
add_to_lists(max_odd, efficient_invariants, all_invariants)
1543
1544
# The number of vertices at odd distance from v and return the min over all vertices
1545
def min_odd(g):
1546
from sage.graphs.distances_all_pairs import distances_all_pairs
1547
D = distances_all_pairs(g)
1548
odds_list = []
1549
for u in D:
1550
odds = 0
1551
for v in D[u]:
1552
if D[u][v] % 2 != 0:
1553
odds += 1
1554
odds_list.append(odds)
1555
return min(odds_list)
1556
add_to_lists(min_odd, efficient_invariants, all_invariants)
1557
1558
#returns sum of distances between *distinct* vertices, return infinity is graph is not connected
1559
def transmission(g):
1560
if not g.is_connected():
1561
return Infinity
1562
if g.is_tree() and max(g.degree()) == 2:
1563
summation = 0
1564
for i in range(1,g.order()):
1565
summation += (i*(i+1))/2
1566
return summation * 2
1567
else:
1568
V = g.vertices()
1569
D = g.distance_all_pairs()
1570
return sum([D[v][w] for v in V for w in V if v != w])
1571
add_to_lists(transmission, efficient_invariants, all_invariants)
1572
1573
def harmonic_index(g):
1574
sum = 0
1575
for (u,v) in g.edges(labels = false):
1576
sum += (2 / (g.degree(u) + g.degree(v)))
1577
return sum
1578
add_to_lists(harmonic_index, efficient_invariants, all_invariants)
1579
1580
def bavelas_index(g):
1581
"""
1582
returns sum over all edges (v,w) of (distance from v to all other vertices)/(distance from w to all other vertices)
1583
computes each edge twice (once with v computation in numerator, once with w computation in numerator)
1584
1585
sage: bavelas_index(p6)
1586
5086/495
1587
sage: bavelas_index(k4)
1588
12
1589
"""
1590
D = g.distance_all_pairs()
1591
1592
def s_aux(v):
1593
"""
1594
computes sum of distances from v to all other vertices
1595
"""
1596
sum = 0
1597
for w in g.vertices():
1598
sum += D[v][w]
1599
return sum
1600
1601
sum_final = 0
1602
1603
for edge in g.edges(labels=false):
1604
v = edge[0]
1605
w = edge[1]
1606
sum_final += (s_aux(w) / s_aux(v)) + (s_aux(v) / s_aux(w))
1607
1608
return sum_final
1609
add_to_lists(bavelas_index, efficient_invariants, all_invariants)
1610
1611
#a solution of the invariant interpolation problem for upper bound of chromatic number for c8chords
1612
#all upper bounds in theory have value at least 3 for c8chords
1613
#returns 2 for bipartite graphs, order for non-bipartite
1614
def bipartite_chromatic(g):
1615
if g.is_bipartite():
1616
return 2
1617
else:
1618
return g.order()
1619
add_to_lists(bipartite_chromatic, efficient_invariants, all_invariants)
1620
1621
def beauchamp_index(g):
1622
"""
1623
Defined on page 597 of Sabidussi, Gert. "The centrality index of a graph." Psychometrika 31.4 (1966): 581-603.
1624
1625
sage: beauchamp_index(c4)
1626
1
1627
sage: beauchamp_index(p5)
1628
137/210
1629
sage: beauchamp_index(graphs.PetersenGraph())
1630
2/3
1631
"""
1632
1633
D = g.distance_all_pairs()
1634
1635
def s_aux(v): #computes sum of distances from v to all other vertices
1636
sum = 0
1637
for w in g.vertices():
1638
sum += D[v][w]
1639
return sum
1640
1641
sum_final = 0
1642
1643
for v in g.vertices():
1644
sum_final += 1/s_aux(v)
1645
return sum_final
1646
1647
add_to_lists(beauchamp_index, efficient_invariants, all_invariants)
1648
1649
def subcubic_tr(g):
1650
"""
1651
Returns the maximum number of vertex disjoint triangles of the graph
1652
1653
Harant, Jochen, et al. "The independence number in graphs of maximum degree three." Discrete Mathematics 308.23 (2008): 5829-5833.
1654
"""
1655
return len(form_triangles_graph(g).connected_components())
1656
add_to_lists(subcubic_tr, efficient_invariants, all_invariants)
1657
1658
def edge_clustering_centrality(g, edge = None):
1659
"""
1660
Returns edge clustering centrality for all edges in a list, or a single centrality for the given edge
1661
Utility to be used with min, avg, max invariants
1662
INPUT: g - a graph
1663
edge - (default: None) An edge in g. If given, will compute centrality for given edge, otherwise all edges. See Graph.has_Edge for acceptable input.
1664
From:
1665
An Application of Edge Clustering Centrality to Brain Connectivity by Joy Lind, Frank Garcea, Bradford Mahon, Roger Vargas, Darren A. Narayan
1666
1667
TESTS:
1668
sage: edge_clustering_centrality(graphs.CompleteGraph(5))
1669
[5, 5, 5, 5, 5, 5, 5, 5, 5, 5]
1670
sage: edge_clustering_centrality(graphs.CompleteBipartiteGraph(3,4))
1671
[2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]
1672
sage: edge_clustering_centrality(graphs.PetersenGraph())
1673
[2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]
1674
sage: edge_clustering_centrality(graphs.BullGraph())
1675
[3, 3, 3, 2, 2]
1676
"""
1677
if edge is None:
1678
edge_clusering_centralities = []
1679
for e in g.edges(labels = False):
1680
edge_clusering_centralities.append(len(set(g.neighbors(e[0])) & set(g.neighbors(e[1]))) + 2) # +2 for the two vertices in e
1681
return edge_clusering_centralities
1682
else:
1683
return len(set(g.neighbors(edge[0])) & set(g.neighbors(edge[1]))) + 2 # +2 for the two vertices in e
1684
1685
def max_edge_clustering_centrality(g):
1686
"""
1687
sage: max_edge_clustering_centrality(p3)
1688
2
1689
sage: max_edge_clustering_centrality(paw)
1690
3
1691
"""
1692
return max(edge_clustering_centrality(g))
1693
add_to_lists(max_edge_clustering_centrality, efficient_invariants, all_invariants)
1694
1695
def min_edge_clustering_centrality(g):
1696
"""
1697
sage: min_edge_clustering_centrality(p3)
1698
2
1699
sage: min_edge_clustering_centrality(paw)
1700
2
1701
"""
1702
return min(edge_clustering_centrality(g))
1703
add_to_lists(min_edge_clustering_centrality, efficient_invariants, all_invariants)
1704
1705
def mean_edge_clustering_centrality(g):
1706
"""
1707
sage: mean_edge_clustering_centrality(p3)
1708
2
1709
sage: mean_edge_clustering_centrality(paw)
1710
11/4
1711
"""
1712
centralities = edge_clustering_centrality(g)
1713
return sum(centralities) / len(centralities)
1714
add_to_lists(mean_edge_clustering_centrality, efficient_invariants, all_invariants)
1715
1716
def local_density(g, vertex = None):
1717
"""
1718
Returns local density for all vertices as a list, or a single local density for the given vertex
1719
INPUT: g - a graph
1720
vertex - (default: None) A vertex in g. If given, it will compute local density for just that vertex, otherwise for all of them
1721
1722
Pavlopoulos, Georgios A., et al. "Using graph theory to analyze biological networks." BioData mining 4.1 (2011): 10.
1723
"""
1724
if vertex == None:
1725
densities = []
1726
for v in g.vertices():
1727
densities.append(g.subgraph(g[v] + [v]).density())
1728
return densities
1729
return g.subgraph(g[vertex] + [vertex]).density()
1730
1731
def min_local_density(g):
1732
"""
1733
sage: min_local_density(p3)
1734
2/3
1735
sage: min_local_density(paw)
1736
2/3
1737
"""
1738
return min(local_density(g))
1739
add_to_lists(min_local_density, efficient_invariants, all_invariants)
1740
1741
def max_local_density(g):
1742
"""
1743
sage: max_local_density(p3)
1744
1
1745
sage: max_local_density(paw)
1746
1
1747
"""
1748
return max(local_density(g))
1749
add_to_lists(max_local_density, efficient_invariants, all_invariants)
1750
1751
def mean_local_density(g):
1752
"""
1753
sage: mean_local_density(p3)
1754
8/9
1755
sage: mean_local_density(paw)
1756
11/12
1757
"""
1758
densities = local_density(g)
1759
return sum(densities) / len(densities)
1760
add_to_lists(mean_local_density, efficient_invariants, all_invariants)
1761
1762
def card_simple_blocks(g):
1763
"""
1764
returns the number of blocks with order 2
1765
1766
sage: card_simple_blocks(k10)
1767
0
1768
sage: card_simple_blocks(paw)
1769
1
1770
sage: card_simple_blocks(kite_with_tail)
1771
1
1772
"""
1773
blocks = g.blocks_and_cut_vertices()[0]
1774
count = 0
1775
for block in blocks:
1776
if len(block) == 2:
1777
count += 1
1778
return count
1779
add_to_lists(card_simple_blocks, efficient_invariants, all_invariants)
1780
1781
# Block of more than 2 vertices
1782
def card_complex_blocks(g):
1783
"""
1784
returns the number of blocks with order 2
1785
1786
sage: card_complex_blocks(k10)
1787
1
1788
sage: card_complex_blocks(paw)
1789
1
1790
sage: card_complex_blocks(kite_with_tail)
1791
1
1792
"""
1793
blocks = g.blocks_and_cut_vertices()[0]
1794
count = 0
1795
for block in blocks:
1796
if len(block) > 2:
1797
count += 1
1798
return count
1799
add_to_lists(card_complex_blocks, efficient_invariants, all_invariants)
1800
1801
# Block is a clique and more than 2 vertices
1802
def card_complex_cliques(g):
1803
"""
1804
returns the number of blocks with order 2
1805
1806
sage: card_complex_cliques(k10)
1807
1
1808
sage: card_complex_cliques(paw)
1809
1
1810
sage: card_complex_cliques(kite_with_tail)
1811
0
1812
"""
1813
blocks = g.blocks_and_cut_vertices()[0]
1814
count = 0
1815
for block in blocks:
1816
h = g.subgraph(block)
1817
if h.is_clique() and h.order() > 2:
1818
count += 1
1819
return count
1820
add_to_lists(card_complex_cliques, efficient_invariants, all_invariants)
1821
1822
def max_minus_min_degrees(g):
1823
return max_degree(g) - min_degree(g)
1824
add_to_lists(max_minus_min_degrees, efficient_invariants, all_invariants)
1825
1826
def randic_irregularity(g):
1827
return order(g)/2 - randic(g)
1828
add_to_lists(randic_irregularity, efficient_invariants, all_invariants)
1829
1830
def degree_variance(g):
1831
avg_degree = g.average_degree()
1832
return 1/order(g) * sum([d**2 - avg_degree for d in [g.degree(v) for v in g.vertices()]])
1833
add_to_lists(degree_variance, efficient_invariants, all_invariants)
1834
1835
def sum_edges_degree_difference(g):
1836
return sum([abs(g.degree(e[0]) - g.degree(e[1])) for e in g.edges()])
1837
add_to_lists(sum_edges_degree_difference, efficient_invariants, all_invariants)
1838
1839
def one_over_size_sedd(g):
1840
return 1/g.size() * sum_edges_degree_difference(g)
1841
add_to_lists(one_over_size_sedd, efficient_invariants, all_invariants)
1842
1843
def largest_eigenvalue_minus_avg_degree(g):
1844
return max_eigenvalue(g) - g.average_degree()
1845
add_to_lists(largest_eigenvalue_minus_avg_degree, efficient_invariants, all_invariants)
1846
1847
def min_betweenness_centrality(g):
1848
centralities = g.centrality_betweenness(exact=True)
1849
return centralities[min(centralities)]
1850
add_to_lists(min_betweenness_centrality, efficient_invariants, all_invariants)
1851
1852
def max_betweenness_centrality(g):
1853
centralities = g.centrality_betweenness(exact=True)
1854
return centralities[max(centralities)]
1855
add_to_lists(max_betweenness_centrality, efficient_invariants, all_invariants)
1856
1857
def mean_betweenness_centrality(g):
1858
centralities = g.centrality_betweenness(exact=True)
1859
return sum([centralities[vertex] for vertex in g.vertices()]) / g.order()
1860
add_to_lists(mean_betweenness_centrality, efficient_invariants, all_invariants)
1861
1862
def min_centrality_closeness(g):
1863
centralities = g.centrality_closeness()
1864
return centralities[min(centralities)]
1865
add_to_lists(min_centrality_closeness, efficient_invariants, all_invariants)
1866
1867
def max_centrality_closeness(g):
1868
centralities = g.centrality_closeness()
1869
return centralities[max(centralities)]
1870
add_to_lists(max_centrality_closeness, efficient_invariants, all_invariants)
1871
1872
def mean_centrality_closeness(g):
1873
centralities = g.centrality_closeness()
1874
return sum([centralities[vertex] for vertex in g.vertices()]) / g.order()
1875
add_to_lists(mean_centrality_closeness, efficient_invariants, all_invariants)
1876
1877
def min_centrality_degree(g):
1878
centralities = g.centrality_degree()
1879
return centralities[min(centralities)]
1880
add_to_lists(min_centrality_degree, efficient_invariants, all_invariants)
1881
1882
def max_centrality_degree(g):
1883
centralities = g.centrality_degree()
1884
return centralities[max(centralities)]
1885
add_to_lists(max_centrality_degree, efficient_invariants, all_invariants)
1886
1887
def mean_centrality_degree(g):
1888
centralities = g.centrality_degree()
1889
return sum([centralities[vertex] for vertex in g.vertices()]) / g.order()
1890
add_to_lists(mean_centrality_degree, efficient_invariants, all_invariants)
1891
1892
def homo_lumo_gap(g):
1893
order = g.order()
1894
if order % 2 != 0:
1895
return 0
1896
eigenvalues = g.spectrum()
1897
# Minus 1 accounts for the 0 indexing of a list
1898
return eigenvalues[floor((order+1)/2) - 1] - eigenvalues[ceil((order+1)/2) - 1]
1899
add_to_lists(homo_lumo_gap, efficient_invariants, all_invariants)
1900
1901
def homo_lumo_index(g):
1902
order = g.order()
1903
eigenvalues = g.adjacency_matrix(sparse=False).change_ring(RDF).eigenvalues(algorithm="symmetric")
1904
if order%2 == 0:
1905
# Minus 1 accounts for the 0 indexing of a list
1906
return max(abs(eigenvalues[floor((order+1)/2) - 1]), abs(eigenvalues[ceil((order+1)/2) - 1]))
1907
else:
1908
return eigenvalues[floor(order/2)]
1909
add_to_lists(homo_lumo_index, efficient_invariants, all_invariants)
1910
1911
def neighborhood_union_nonadjacent(g):
1912
# Define that for copmlete graphs (i.e. nothing to minimize over later), return n, which is trivial upper bound.
1913
all_dist = g.distance_all_pairs()
1914
nonadj = [(v,w) for v in g for w in g if all_dist[v][w] > 1]
1915
if not nonadj:
1916
return g.order()
1917
else:
1918
return min( len(union(g.neighbors(v), g.neighbors(w))) for (v,w) in nonadj)
1919
add_to_lists(neighborhood_union_nonadjacent, efficient_invariants, all_invariants)
1920
1921
def neighborhood_union_dist2(g):
1922
# Define that for graphs with no dist 2 (i.e. nothing to minimize over later), return n, which is trivial upper bound.
1923
all_dist = g.distance_all_pairs()
1924
dist2 = [(v,w) for v in g for w in g if all_dist[v][w] == 2]
1925
if not dist2:
1926
return g.order()
1927
else:
1928
return min( len(union(g.neighbors(v), g.neighbors(w))) for (v, w) in dist2)
1929
add_to_lists(neighborhood_union_dist2, efficient_invariants, all_invariants)
1930
1931
def simplical_vertices(g):
1932
"""
1933
The number of simplical vertices in g.
1934
v is simplical if the induced nieghborhood is a clique.
1935
"""
1936
return sum( is_simplical_vertex(g,v) for v in g.vertices() )
1937
add_to_lists(simplical_vertices, efficient_invariants, all_invariants)
1938
1939
def first_zagreb_index(g):
1940
"""
1941
The sume of squares of the degrees
1942
"""
1943
return sum(g.degree(v)**2 for v in g.vertices())
1944
add_to_lists(first_zagreb_index, efficient_invariants, all_invariants)
1945
1946
def degree_two_vertices(g):
1947
"""
1948
The number of degree 2 vertices
1949
"""
1950
return len([deg for deg in g.degree() if deg == 2])
1951
add_to_lists(degree_two_vertices, efficient_invariants, all_invariants)
1952
1953
def degree_order_minus_one_vertices(g):
1954
"""
1955
The number of vertices with degree = n-1
1956
"""
1957
return len([deg for deg in g.degree() if deg == g.order() - 1])
1958
add_to_lists(degree_order_minus_one_vertices, efficient_invariants, all_invariants)
1959
1960
def maximum_degree_vertices(g):
1961
"""
1962
The number of vertices with degree equal to the maximum degree
1963
"""
1964
return len([deg for deg in g.degree() if deg == max_degree(g)])
1965
add_to_lists(maximum_degree_vertices, efficient_invariants, all_invariants)
1966
1967
def minimum_degree_vertices(g):
1968
"""
1969
The number of vertices with degree equal to the minimum degree
1970
"""
1971
return len([deg for deg in g.degree() if deg == min_degree(g)])
1972
add_to_lists(minimum_degree_vertices, efficient_invariants, all_invariants)
1973
1974
def second_zagreb_index(g):
1975
"""
1976
The sum over all edges (v,w) of the product of degrees(v)*degree(w)
1977
"""
1978
return sum(g.degree(v)*g.degree(w) for (v,w) in g.edge_iterator(labels=False))
1979
add_to_lists(second_zagreb_index, efficient_invariants, all_invariants)
1980
1981
# Damir Vukicevic, Qiuli Li, Jelena Sedlar, and Tomislav Doslic, Lanzhou Index. MATCH Commun. Math. Comput. Chem., 80: 863-876, 2018.
1982
def lanzhou_index(g):
1983
"""
1984
The sum over all vertices v of products of the co-degree of v (deg(v) in the complement of g) times the square of deg(v).
1985
1986
sage: lanzhou_index(graphs.CompleteGraph(10))
1987
0
1988
sage: lanzhou_index(graphs.CompleteBipartiteGraph(5,5))
1989
1000
1990
"""
1991
n = g.order()
1992
return sum( ((n-1) - g.degree(v)) * (g.degree(v) ** 2) for v in g.vertices() )
1993
add_to_lists(lanzhou_index, efficient_invariants, all_invariants)
1994
1995
def friendship_number(g):
1996
"""
1997
The friendship number of a graph is the number of pairs of vertices that have a unique common neighbour.
1998
1999
sage: friendship_number(graphs.FriendshipGraph(3))
2000
21
2001
sage: friendship_number(graphs.CompleteGraph(7))
2002
0
2003
"""
2004
from itertools import combinations
2005
return sum((1 if len(common_neighbors(g, u, v))==1 else 0) for (u,v) in combinations(g.vertices(), 2))
2006
add_to_lists(friendship_number, efficient_invariants, all_invariants)
2007
2008
#####
2009
# INTRACTABLE INVATIANTS
2010
#####
2011
def domination_number(g):
2012
"""
2013
Returns the domination number of the graph g, i.e., the size of a maximum
2014
dominating set.
2015
2016
A complete graph is dominated by any of its vertices::
2017
2018
sage: domination_number(graphs.CompleteGraph(5))
2019
1
2020
2021
A star graph is dominated by its central vertex::
2022
2023
sage: domination_number(graphs.StarGraph(5))
2024
1
2025
2026
The domination number of a cycle of length n is the ceil of n/3.
2027
2028
sage: domination_number(graphs.CycleGraph(5))
2029
2
2030
"""
2031
return g.dominating_set(value_only=True)
2032
add_to_lists(domination_number, intractable_invariants, all_invariants)
2033
2034
def independence_number(g):
2035
return g.independent_set(value_only=True)
2036
add_to_lists(independence_number, intractable_invariants, all_invariants)
2037
2038
def clique_covering_number(g):
2039
# Finding the chromatic number of the complement of a fullerene
2040
# is extremely slow, even when using MILP as the algorithm.
2041
# Therefore we check to see if the graph is triangle-free.
2042
# If it is, then the clique covering number is equal to the
2043
# number of vertices minus the size of a maximum matching.
2044
if g.is_triangle_free():
2045
return g.order() - matching_number(g)
2046
gc = g.complement()
2047
return gc.chromatic_number(algorithm="MILP")
2048
add_to_lists(clique_covering_number, intractable_invariants, all_invariants)
2049
2050
def n_over_alpha(g):
2051
n = g.order() + 0.0
2052
return n/independence_number(g)
2053
add_to_lists(n_over_alpha, intractable_invariants, all_invariants)
2054
2055
def independent_dominating_set_number(g):
2056
return g.dominating_set(value_only=True, independent=True)
2057
add_to_lists(independent_dominating_set_number, intractable_invariants, all_invariants)
2058
2059
# Clearly intractable
2060
# alpha / order
2061
def independence_ratio(g):
2062
return independence_number(g)/(g.order()+0.0)
2063
add_to_lists(independence_ratio, intractable_invariants, all_invariants)
2064
2065
def min_degree_of_max_ind_set(g):
2066
"""
2067
Returns the minimum degree of any vertex that is a part of any maximum indepdendent set
2068
2069
sage: min_degree_of_max_ind_set(c4)
2070
2
2071
sage: min_degree_of_max_ind_set(graphs.PetersenGraph())
2072
3
2073
"""
2074
2075
low_degree = g.order()
2076
list_of_vertices = []
2077
2078
UnionSet = Set({})
2079
IndSets = find_all_max_ind_sets(g)
2080
2081
for s in IndSets:
2082
UnionSet = UnionSet.union(Set(s))
2083
2084
list_of_vertices = list(UnionSet)
2085
2086
for v in list_of_vertices:
2087
if g.degree(v) < low_degree:
2088
low_degree = g.degree(v)
2089
2090
return low_degree
2091
add_to_lists(min_degree_of_max_ind_set, intractable_invariants, all_invariants)
2092
2093
def bipartite_number(g):
2094
"""
2095
Defined as the largest number of vertices that induces a bipartite subgraph
2096
2097
sage: bipartite_number(graphs.PetersenGraph())
2098
7
2099
sage: bipartite_number(c4)
2100
4
2101
sage: bipartite_number(graphs.CompleteGraph(3))
2102
2
2103
"""
2104
if g.is_bipartite():
2105
return g.order()
2106
return len(max_bipartite_set(g, [], g.vertices()))
2107
add_to_lists(bipartite_number, intractable_invariants, all_invariants)
2108
2109
# Needs Enhancement
2110
def edge_bipartite_number(g):
2111
"""
2112
Defined as the largest number of edges in an induced bipartite subgraph
2113
2114
sage: edge_bipartite_number(graphs.CompleteGraph(5))
2115
1
2116
sage: edge_bipartite_number(graphs.CompleteBipartiteGraph(5, 5))
2117
25
2118
sage: edge_bipartite_number(graphs.ButterflyGraph())
2119
2
2120
"""
2121
return g.subgraph(max_bipartite_set(g, [], g.vertices())).size()
2122
add_to_lists(edge_bipartite_number, intractable_invariants, all_invariants)
2123
2124
def cheeger_constant(g):
2125
"""
2126
Defined at https://en.wikipedia.org/wiki/Cheeger_constant_(graph_theory)
2127
2128
sage: cheeger_constant(graphs.PathGraph(2))
2129
1
2130
sage: cheeger_constant(graphs.CompleteGraph(5))
2131
3
2132
sage: cheeger_constant(paw)
2133
1
2134
"""
2135
n = g.order()
2136
upper = floor(n/2)
2137
2138
v = g.vertices()
2139
SetV = Set(v)
2140
2141
temp = g.order()
2142
best = n
2143
2144
for i in range(1, upper+1):
2145
for s in SetV.subsets(i):
2146
count = 0
2147
for u in s:
2148
for w in SetV.difference(s):
2149
for e in g.edges(labels=false):
2150
if Set([u,w]) == Set(e):
2151
count += 1
2152
temp = count/i
2153
if temp < best:
2154
best = temp
2155
return best
2156
add_to_lists(cheeger_constant, intractable_invariants, all_invariants)
2157
2158
def tr(g):
2159
"""
2160
Returns the maximum number of vertex disjoint triangles of the graph
2161
2162
Harant, Jochen, et al. "The independence number in graphs of maximum degree three." Discrete Mathematics 308.23 (2008): 5829-5833.
2163
"""
2164
if is_subcubic(g):
2165
return subcubic_tr(g)
2166
return independence_number(form_triangles_graph(g))
2167
add_to_lists(tr, intractable_invariants, all_invariants)
2168
2169
def total_domination_number(g):
2170
return g.dominating_set(total=True, value_only=True)
2171
add_to_lists(total_domination_number, intractable_invariants, all_invariants)
2172
2173
# A graph G is t-tough for real t if for every integer k>1, G cannot be split into k connected components by removal of fewer than tk vertices
2174
# Returns Infinity if g is complete
2175
# Inefficient to calculate
2176
def toughness(g):
2177
"""
2178
Tests:
2179
sage: toughness(graphs.PathGraph(3))
2180
0.5
2181
sage: toughness(graphs.CompleteGraph(5))
2182
+Infinity
2183
sage: toughness(graphs.PetersenGraph())
2184
1.3333333333333333
2185
"""
2186
order = g.order()
2187
t = Infinity
2188
for x in Subsets(g.vertices()):
2189
if x and len(x) != order: # Proper, non-empty subset
2190
H = copy(g)
2191
H.delete_vertices(x)
2192
k = H.connected_components_number()
2193
if k > 1:
2194
t = min(float(len(x)) / k, t)
2195
return t
2196
add_to_lists(toughness, intractable_invariants, all_invariants)
2197
2198
# Sigma_k = min( sum( degrees(v_i) : every k-element independent set v_1,..,v_k ) )
2199
# Inefficient to calculate
2200
def sigma_k(g,k):
2201
"""
2202
Tests:
2203
sage: sigma_k(graphs.CompleteGraph(5), 1)
2204
4
2205
sage: sigma_k(graphs.PathGraph(4), 2)
2206
2
2207
"""
2208
sigma = Infinity
2209
for S in Subsets(g.vertices(), k):
2210
if g.is_independent_set(S):
2211
sigma = min(sigma, sum([g.degree(x) for x in S]) )
2212
return sigma
2213
2214
def sigma_2(g):
2215
return sigma_k(g,2)
2216
def sigma_3(g):
2217
return sigma_k(g,3)
2218
add_to_lists(sigma_2, intractable_invariants, all_invariants)
2219
add_to_lists(sigma_3, intractable_invariants, all_invariants)
2220
2221
def homogenous_number(g):
2222
"""
2223
Equals the larger of the independence number or the clique number
2224
"""
2225
return max(independence_number(g), g.clique_number())
2226
add_to_lists(homogenous_number, intractable_invariants, all_invariants)
2227
2228
def edge_domination_number(g):
2229
"""
2230
The minimum size of a set of edges S such that every edge not in S is incident to an edge in S
2231
"""
2232
return domination_number(g.line_graph())
2233
add_to_lists(edge_domination_number, intractable_invariants, all_invariants)
2234
2235
def circumference(g):
2236
"""
2237
Returns length of longest cycle in g
2238
2239
If acyclic, throws a ValueError. Some define this to be 0; we leave it up to the user.
2240
"""
2241
lengths = cycle_lengths(g)
2242
if not lengths:
2243
raise ValueError("Graph is acyclic. Circumference undefined")
2244
else:
2245
return max(lengths)
2246
add_to_lists(circumference, intractable_invariants, all_invariants)
2247
2248
def tree_number(g):
2249
"""
2250
The order of a maximum-size induced subgraph that's a tree in g
2251
2252
See Erdös, Paul, Michael Saks, and Vera T. Sós. "Maximum induced trees in graphs." Journal of Combinatorial Theory, Series B 41.1 (1986): 61-79.
2253
"""
2254
return max_induced_tree(g).order()
2255
add_to_lists(tree_number, intractable_invariants, all_invariants)
2256
2257
def forest_number(g):
2258
"""
2259
The order of a maximum-size induced subgraph of g that's a forest
2260
"""
2261
return max_induced_forest(g).order()
2262
add_to_lists(forest_number, intractable_invariants, all_invariants)
2263
2264
def minimum_maximal_matching_size(g):
2265
"""
2266
The minimum number of edges k s.t. there exists a matching of size k which is not extendable
2267
"""
2268
if(g.size() == 0):
2269
return 0
2270
2271
matchings_old = []
2272
matchings = [[e] for e in g.edges()]
2273
while True:
2274
matchings_old = matchings
2275
matchings = []
2276
for matching in matchings_old:
2277
extendable = False
2278
for e in (edge for edge in g.edges() if edge not in matching):
2279
possible_matching = matching + [e]
2280
if is_matching(possible_matching):
2281
matchings.append(possible_matching)
2282
extendable = True
2283
if not extendable:
2284
return len(matching)
2285
add_to_lists(minimum_maximal_matching_size, intractable_invariants, all_invariants)
2286
2287
def hamiltonian_index(g):
2288
"""
2289
Returns i, where L^i(g) = L(L(L(...L(g)))) is the first line graph iterate of g such that L^i(g) is Hamiltonian
2290
2291
If g is Hamiltonian, then h(G) = 0.
2292
Raises ValueError if g is disconnected or if g is a simple path, since h(g) is undefined for either.
2293
2294
Defined in: Chartrand, Gary. "On hamiltonian line-graphs." Transactions of the American Mathematical Society 134.3 (1968): 559-566.
2295
2296
sage: hamiltonian_index(graphs.CycleGraph(5))
2297
0
2298
sage: hamiltonian_index(graphs.PetersenGraph())
2299
1
2300
sage: hamiltonian_index(graphs.TadpoleGraph(4, 3))
2301
3
2302
"""
2303
if not g.is_connected():
2304
raise ValueError("The input graph g is not connected. The Hamiltonian index is only defined for connected graphs.")
2305
if g.is_isomorphic(graphs.PathGraph(g.order())):
2306
raise ValueError("The input graph g is a simple path. The Hamiltonian index is not defined for path graphs.")
2307
line_graph_i = g
2308
for index in xrange(0, (g.order() - 3) + 1): # [Chartrand, 68] proved index is upper bounded by n - 3.
2309
if line_graph_i.is_hamiltonian():
2310
return index
2311
line_graph_i = line_graph_i.line_graph()
2312
add_to_lists(hamiltonian_index, intractable_invariants, all_invariants)
2313
2314
2315
#FAST ENOUGH (tested for graphs on 140921): lovasz_theta, clique_covering_number, all efficiently_computable
2316
#SLOW but FIXED for SpecialGraphs
2317
2318
print("loaded invariants")
2319
2320
#############################################################################
2321
# End of invariants section #
2322
#############################################################################
2323
# GRAPH PROPERTIES
2324
2325
def has_star_center(g):
2326
"""
2327
Evalutes whether graph ``g`` has a vertex adjacent to all others.
2328
2329
EXAMPLES:
2330
2331
sage: has_star_center(flower_with_3_petals)
2332
True
2333
2334
sage: has_star_center(c4)
2335
False
2336
2337
Edge cases ::
2338
2339
sage: has_star_center(Graph(1))
2340
True
2341
2342
sage: has_star_center(Graph(0))
2343
False
2344
"""
2345
return (g.order() - 1) in g.degree()
2346
2347
def is_complement_of_chordal(g):
2348
"""
2349
Evaluates whether graph ``g`` is a complement of a chordal graph.
2350
2351
A chordal graph is one in which all cycles of four or more vertices have a
2352
chord, which is an edge that is not part of the cycle but connects two
2353
vertices of the cycle.
2354
2355
EXAMPLES:
2356
2357
sage: is_complement_of_chordal(p4)
2358
True
2359
2360
sage: is_complement_of_chordal(Graph(4))
2361
True
2362
2363
sage: is_complement_of_chordal(p5)
2364
False
2365
2366
Any graph without a 4-or-more cycle is vacuously chordal. ::
2367
2368
sage: is_complement_of_chordal(graphs.CompleteGraph(4))
2369
True
2370
2371
sage: is_complement_of_chordal(Graph(3))
2372
True
2373
2374
sage: is_complement_of_chordal(Graph(0))
2375
True
2376
"""
2377
return g.complement().is_chordal()
2378
2379
def pairs_have_unique_common_neighbor(g):
2380
"""
2381
Evalaute if each pair of vertices in ``g`` has exactly one common neighbor.
2382
2383
Also known as the friendship property.
2384
By the Friendship Theorem, the only connected graphs with the friendship
2385
property are flowers.
2386
2387
EXAMPLES:
2388
2389
sage: pairs_have_unique_common_neighbor(flower(5))
2390
True
2391
2392
sage: pairs_have_unique_common_neighbor(k3)
2393
True
2394
2395
sage: pairs_have_unique_common_neighbor(k4)
2396
False
2397
2398
sage: pairs_have_unique_common_neighbor(graphs.CompleteGraph(2))
2399
False
2400
2401
Vacuous cases ::
2402
2403
sage: pairs_have_unique_common_neighbor(Graph(1))
2404
True
2405
2406
sage: pairs_have_unique_common_neighbor(Graph(0))
2407
True
2408
"""
2409
from itertools import combinations
2410
for (u,v) in combinations(g.vertices(), 2):
2411
if len(common_neighbors(g, u, v)) != 1:
2412
return False
2413
return True
2414
2415
def is_distance_transitive(g):
2416
"""
2417
Evaluates if graph ``g`` is distance transitive.
2418
2419
A graph is distance transitive if all a,b,u,v satisfy that
2420
dist(a,b) = dist(u,v) implies there's an automorphism with a->u and b->v.
2421
2422
EXAMPLES:
2423
2424
sage: is_distance_transitive(graphs.CompleteGraph(4))
2425
True
2426
2427
sage: is_distance_transitive(graphs.PetersenGraph())
2428
True
2429
2430
sage: is_distance_transitive(Graph(3))
2431
True
2432
2433
sage: is_distance_transitive(graphs.ShrikhandeGraph())
2434
False
2435
2436
This method accepts disconnected graphs. ::
2437
2438
sage: is_distance_transitive(graphs.CompleteGraph(3).disjoint_union(graphs.CompleteGraph(3)))
2439
True
2440
2441
sage: is_distance_transitive(graphs.CompleteGraph(2).disjoint_union(Graph(2)))
2442
False
2443
2444
Vacuous cases ::
2445
2446
sage: is_distance_transitive(Graph(0))
2447
True
2448
2449
sage: is_distance_transitive(Graph(1))
2450
True
2451
2452
sage: is_distance_transitive(Graph(2))
2453
True
2454
2455
... WARNING ::
2456
2457
This method calls, via the automorphism group, the Gap package. This
2458
package behaves badly with most threading or multiprocessing tools.
2459
"""
2460
from itertools import combinations
2461
dist_dict = g.distance_all_pairs()
2462
auto_group = g.automorphism_group()
2463
2464
for d in g.distances_distribution():
2465
sameDistPairs = []
2466
for (u,v) in combinations(g.vertices(), 2):
2467
# By default, no entry if disconnected. We substitute +Infinity.
2468
if dist_dict[u].get(v, +Infinity) == d:
2469
sameDistPairs.append(Set([u,v]))
2470
if len(sameDistPairs) >= 2:
2471
if len(sameDistPairs) != len(auto_group.orbit(sameDistPairs[0], action = "OnSets")):
2472
return False
2473
return True
2474
2475
def is_dirac(g):
2476
"""
2477
Evaluates if graph ``g`` has order at least 3 and min. degree at least n/2.
2478
2479
See Dirac's Theorem: If graph is_dirac, then it is hamiltonian.
2480
2481
EXAMPLES:
2482
2483
sage: is_dirac(graphs.CompleteGraph(6))
2484
True
2485
2486
sage: is_dirac(graphs.CompleteGraph(3))
2487
True
2488
2489
sage: is_dirac(graphs.CompleteGraph(2))
2490
False
2491
2492
sage: is_dirac(graphs.CycleGraph(5))
2493
False
2494
"""
2495
n = g.order()
2496
return n > 2 and min(g.degree()) >= n/2
2497
2498
def is_ore(g):
2499
"""
2500
Evaluate if deg(v)+deg(w)>=n for all non-adjacent pairs v,w in graph ``g``.
2501
2502
See Ore's Theorem: If graph is_ore, then it is hamiltonian.
2503
2504
EXAMPLES:
2505
2506
sage: is_ore(graphs.CompleteGraph(5))
2507
True
2508
2509
sage: is_ore(graphs.CompleteGraph(2))
2510
True
2511
2512
sage: is_ore(dart)
2513
False
2514
2515
sage: is_ore(Graph(2))
2516
False
2517
2518
sage: is_ore(graphs.CompleteGraph(2).disjoint_union(Graph(1)))
2519
False
2520
2521
Vacous cases ::
2522
2523
sage: is_ore(Graph(0))
2524
True
2525
2526
sage: is_ore(Graph(1))
2527
True
2528
"""
2529
A = g.adjacency_matrix()
2530
n = g.order()
2531
D = g.degree()
2532
for i in xrange(n):
2533
for j in xrange(i):
2534
if A[i][j]==0:
2535
if D[i] + D[j] < n:
2536
return False
2537
return True
2538
2539
def is_haggkvist_nicoghossian(g):
2540
"""
2541
Evaluates if g is 2-connected and min degree >= (n + vertex_connectivity)/3.
2542
2543
INPUT:
2544
2545
- ``g`` -- graph
2546
2547
EXAMPLES:
2548
2549
sage: is_haggkvist_nicoghossian(graphs.CompleteGraph(3))
2550
True
2551
2552
sage: is_haggkvist_nicoghossian(graphs.CompleteGraph(5))
2553
True
2554
2555
sage: is_haggkvist_nicoghossian(graphs.CycleGraph(5))
2556
False
2557
2558
sage: is_haggkvist_nicoghossian(graphs.CompleteBipartiteGraph(4,3)
2559
False
2560
2561
sage: is_haggkvist_nicoghossian(Graph(1))
2562
False
2563
2564
sage: is_haggkvist_nicoghossian(graphs.CompleteGraph(2))
2565
False
2566
2567
REFERENCES:
2568
2569
Theorem: If a graph ``is_haggkvist_nicoghossian``, then it is Hamiltonian.
2570
2571
.. [HN1981] \R. Häggkvist and G. Nicoghossian, "A remark on Hamiltonian
2572
cycles". Journal of Combinatorial Theory, Series B, 30(1):
2573
118--120, 1981.
2574
"""
2575
k = g.vertex_connectivity()
2576
return k >= 2 and min(g.degree()) >= (1.0/3) * (g.order() + k)
2577
2578
def is_genghua_fan(g):
2579
"""
2580
Evaluates if graph ``g`` satisfies a condition for Hamiltonicity by G. Fan.
2581
2582
OUTPUT:
2583
2584
Returns ``True`` if ``g`` is 2-connected and satisfies that
2585
`dist(u,v)=2` implies `\max(deg(u), deg(v)) \geq n/2` for all
2586
vertices `u,v`.
2587
Returns ``False`` otherwise.
2588
2589
EXAMPLES:
2590
2591
sage: is_genghua_fan(graphs.DiamondGraph())
2592
True
2593
2594
sage: is_genghua_fan(graphs.CycleGraph(4))
2595
False
2596
2597
sage: is_genghua_fan(graphs.ButterflyGraph())
2598
False
2599
2600
sage: is_genghua_fan(Graph(1))
2601
False
2602
2603
REFERENCES:
2604
2605
Theorem: If a graph ``is_genghua_fan``, then it is Hamiltonian.
2606
2607
.. [Fan1984] Geng-Hua Fan, "New sufficient conditions for cycles in
2608
graphs". Journal of Combinatorial Theory, Series B, 37(3):
2609
221--227, 1984.
2610
"""
2611
if not is_two_connected(g):
2612
return False
2613
D = g.degree()
2614
Dist = g.distance_all_pairs()
2615
V = g.vertices()
2616
n = g.order()
2617
for i in xrange(n):
2618
for j in xrange(i):
2619
if Dist[V[i]][V[j]] == 2 and max(D[i], D[j]) < n / 2.0:
2620
return False
2621
return True
2622
2623
def is_planar_transitive(g):
2624
"""
2625
Evaluates whether graph ``g`` is planar and is vertex-transitive.
2626
2627
EXAMPLES:
2628
2629
sage: is_planar_transitive(graphs.HexahedralGraph())
2630
True
2631
2632
sage: is_planar_transitive(graphs.CompleteGraph(2))
2633
True
2634
2635
sage: is_planar_transitive(graphs.FranklinGraph())
2636
False
2637
2638
sage: is_planar_transitive(graphs.BullGraph())
2639
False
2640
2641
Vacuous cases ::
2642
2643
sage: is_planar_transitive(Graph(1))
2644
True
2645
2646
Sage defines `Graph(0).is_vertex_transitive() == False``. ::
2647
2648
sage: is_planar_transitive(Graph(0))
2649
False
2650
"""
2651
return g.is_planar() and g.is_vertex_transitive()
2652
2653
def is_generalized_dirac(g):
2654
"""
2655
Test if ``graph`` g meets condition in a generalization of Dirac's Theorem.
2656
2657
OUTPUT:
2658
2659
Returns ``True`` if g is 2-connected and for all non-adjacent u,v,
2660
the cardinality of the union of neighborhood(u) and neighborhood(v)
2661
is `>= (2n-1)/3`.
2662
2663
EXAMPLES:
2664
2665
sage: is_generalized_dirac(graphs.HouseGraph())
2666
True
2667
2668
sage: is_generalized_dirac(graphs.PathGraph(5))
2669
False
2670
2671
sage: is_generalized_dirac(graphs.DiamondGraph())
2672
False
2673
2674
sage: is_generalized_dirac(Graph(1))
2675
False
2676
2677
REFERENCES:
2678
2679
Theorem: If graph g is_generalized_dirac, then it is Hamiltonian.
2680
2681
.. [FGJS1989] \R.J. Faudree, Ronald Gould, Michael Jacobson, and
2682
R.H. Schelp, "Neighborhood unions and hamiltonian
2683
properties in graphs". Journal of Combinatorial
2684
Theory, Series B, 47(1): 1--9, 1989.
2685
"""
2686
from itertools import combinations
2687
2688
if not is_two_connected(g):
2689
return False
2690
for (u,v) in combinations(g.vertices(), 2):
2691
if not g.has_edge(u,v):
2692
if len(neighbors_set(u, v)) < (2.0 * g.order() - 1) / 3:
2693
return False
2694
return True
2695
2696
def is_van_den_heuvel(g):
2697
"""
2698
Evaluates if g meets an eigenvalue condition related to Hamiltonicity.
2699
2700
INPUT:
2701
2702
- ``g`` -- graph
2703
2704
OUTPUT:
2705
2706
Let ``g`` be of order `n`.
2707
Let `A_H` denote the adjacency matrix of a graph `H`, and `D_H` denote
2708
the matrix with the degrees of the vertices of `H` on the diagonal.
2709
Define `Q_H = D_H + A_H` and `L_H = D_H - A_H` (i.e. the Laplacian).
2710
Finally, let `C` be the cycle graph on `n` vertices.
2711
2712
Returns ``True`` if the `i`-th eigenvalue of `L_C` is at most the `i`-th
2713
eigenvalue of `L_g` and the `i`-th eigenvalue of `Q_C` is at most the
2714
`i`-th eigenvalue of `Q_g for all `i`.
2715
2716
EXAMPLES:
2717
2718
sage: is_van_den_heuvel(graphs.CycleGraph(5))
2719
True
2720
2721
sage: is_van_den_heuvel(graphs.PetersenGraph())
2722
False
2723
2724
REFERENCES:
2725
2726
Theorem: If a graph is Hamiltonian, then it ``is_van_den_heuvel``.
2727
2728
.. [Heu1995] \J.van den Heuvel, "Hamilton cycles and eigenvalues of
2729
graphs". Linear Algebra and its Applications, 226--228:
2730
723--730, 1995.
2731
2732
TESTS::
2733
2734
sage: is_van_den_heuvel(Graph(0))
2735
False
2736
2737
sage: is_van_den_heuvel(Graph(1))
2738
True
2739
"""
2740
cycle_n = graphs.CycleGraph(g.order())
2741
2742
cycle_laplac_eigen = sorted(cycle_n.laplacian_matrix().eigenvalues())
2743
g_laplac_eigen = sorted(g.laplacian_matrix().eigenvalues())
2744
for cycle_lambda_i, g_lambda_i in zip(cycle_laplac_eigen, g_laplac_eigen):
2745
if cycle_lambda_i > g_lambda_i:
2746
return False
2747
2748
def Q(g):
2749
A = g.adjacency_matrix(sparse=False)
2750
D = matrix(g.order(), sparse=False)
2751
row_sums = [sum(r) for r in A.rows()]
2752
for i in xrange(A.nrows()):
2753
D[i,i] = row_sums[i]
2754
return D + A
2755
cycle_q_matrix = sorted(Q(cycle_n).eigenvalues())
2756
g_q_matrix = sorted(Q(g).eigenvalues())
2757
for cycle_q_lambda_i, g_q_lambda_i in zip(cycle_q_matrix, g_q_matrix):
2758
if cycle_q_lambda_i > g_q_lambda_i:
2759
return False
2760
2761
return True
2762
2763
def is_two_connected(g):
2764
"""
2765
Evaluates whether graph ``g`` is 2-connected.
2766
2767
A 2-connected graph is a connected graph on at least 3 vertices such that
2768
the removal of any single vertex still gives a connected graph.
2769
Follows convention that complete graph `K_n` is `n-1`-connected.
2770
2771
Almost equivalent to ``Graph.is_biconnected()``. We prefer our name. AND,
2772
while that method defines that ``graphs.CompleteGraph(2)`` is biconnected,
2773
we follow the convention that `K_n` is `n-1`-connected, so `K_2` is
2774
only 1-connected.
2775
2776
EXAMPLES:
2777
2778
sage: is_two_connected(graphs.CycleGraph(5))
2779
True
2780
2781
sage: is_two_connected(graphs.CompleteGraph(3))
2782
True
2783
2784
sage: is_two_connected(graphs.PathGraph(5))
2785
False
2786
2787
sage: is_two_connected(graphs.CompleteGraph(2))
2788
False
2789
2790
sage: is_two_connected(Graph(3))
2791
False
2792
2793
Edge cases ::
2794
2795
sage: is_two_connected(Graph(0))
2796
False
2797
2798
sage: is_two_connected(Graph(1))
2799
False
2800
"""
2801
if g.is_isomorphic(graphs.CompleteGraph(2)):
2802
return False
2803
return g.is_biconnected()
2804
2805
def is_three_connected(g):
2806
"""
2807
Evaluates whether graph ``g`` is 3-connected.
2808
2809
A 3-connected graph is a connected graph on at least 4 vertices such that
2810
the removal of any two vertices still gives a connected graph.
2811
Follows convention that complete graph `K_n` is `n-1`-connected.
2812
2813
EXAMPLES:
2814
2815
sage: is_three_connected(graphs.PetersenGraph())
2816
True
2817
2818
sage: is_three_connected(graphs.CompleteGraph(4))
2819
True
2820
2821
sage: is_three_connected(graphs.CycleGraph(5))
2822
False
2823
2824
sage: is_three_connected(graphs.PathGraph(5))
2825
False
2826
2827
sage: is_three_connected(graphs.CompleteGraph(3))
2828
False
2829
2830
sage: is_three_connected(graphs.CompleteGraph(2))
2831
False
2832
2833
sage: is_three_connected(Graph(4))
2834
False
2835
2836
Edge cases ::
2837
2838
sage: is_three_connected(Graph(0))
2839
False
2840
2841
sage: is_three_connected(Graph(1))
2842
False
2843
2844
.. WARNING::
2845
2846
Implementation requires Sage 8.2+.
2847
"""
2848
return g.vertex_connectivity(k = 3)
2849
2850
def is_four_connected(g):
2851
"""
2852
Evaluates whether ``g`` is 4-connected.
2853
2854
A 4-connected graph is a connected graph on at least 5 vertices such that
2855
the removal of any three vertices still gives a connected graph.
2856
Follows convention that complete graph `K_n` is `n-1`-connected.
2857
2858
EXAMPLES:
2859
2860
2861
sage: is_four_connected(graphs.CompleteGraph(5))
2862
True
2863
2864
sage: is_four_connected(graphs.PathGraph(5))
2865
False
2866
2867
sage: is_four_connected(Graph(5))
2868
False
2869
2870
sage: is_four_connected(graphs.CompleteGraph(4))
2871
False
2872
2873
Edge cases ::
2874
2875
sage: is_four_connected(Graph(0))
2876
False
2877
2878
sage: is_four_connected(Graph(1))
2879
False
2880
2881
.. WARNING::
2882
2883
Implementation requires Sage 8.2+.
2884
"""
2885
return g.vertex_connectivity(k = 4)
2886
2887
def is_lindquester(g):
2888
"""
2889
Test if graph ``g`` meets a neighborhood union condition for Hamiltonicity.
2890
2891
OUTPUT:
2892
2893
Let ``g`` be of order `n`.
2894
2895
Returns ``True`` if ``g`` is 2-connected and for all vertices `u,v`,
2896
`dist(u,v) = 2` implies that the cardinality of the union of
2897
neighborhood(`u`) and neighborhood(`v`) is `\geq (2n-1)/3`.
2898
Returns ``False`` otherwise.
2899
2900
EXAMPLES:
2901
2902
sage: is_lindquester(graphs.HouseGraph())
2903
True
2904
2905
sage: is_lindquester(graphs.OctahedralGraph())
2906
True
2907
2908
sage: is_lindquester(graphs.PathGraph(3))
2909
False
2910
2911
sage: is_lindquester(graphs.DiamondGraph())
2912
False
2913
2914
REFERENCES:
2915
2916
Theorem: If a graph ``is_lindquester``, then it is Hamiltonian.
2917
2918
.. [Lin1989] \T.E. Lindquester, "The effects of distance and
2919
neighborhood union conditions on hamiltonian properties
2920
in graphs". Journal of Graph Theory, 13(3): 335-352,
2921
1989.
2922
"""
2923
if not is_two_connected(g):
2924
return False
2925
D = g.distance_all_pairs()
2926
n = g.order()
2927
V = g.vertices()
2928
for i in range(n):
2929
for j in range(i):
2930
if D[V[i]][V[j]] == 2:
2931
if len(neighbors_set(g,[V[i],V[j]])) < (2*n-1)/3.0:
2932
return False
2933
return True
2934
2935
def is_complete(g):
2936
"""
2937
Tests whether ``g`` is a complete graph.
2938
2939
OUTPUT:
2940
2941
Returns ``True`` if ``g`` is a complete graph; returns ``False`` otherwise.
2942
A complete graph is one where every vertex is connected to every others
2943
vertex.
2944
2945
EXAMPLES:
2946
2947
sage: is_complete(graphs.CompleteGraph(1))
2948
True
2949
2950
sage: is_complete(graphs.CycleGraph(3))
2951
True
2952
2953
sage: is_complete(graphs.CompleteGraph(6))
2954
True
2955
2956
sage: is_complete(Graph(0))
2957
True
2958
2959
sage: is_complete(graphs.PathGraph(5))
2960
False
2961
2962
sage: is_complete(graphs.CycleGraph(4))
2963
False
2964
"""
2965
n = g.order()
2966
e = g.size()
2967
if not g.has_multiple_edges():
2968
return e == n*(n-1)/2
2969
else:
2970
D = g.distance_all_pairs()
2971
for i in range(n):
2972
for j in range(i):
2973
if D[V[i]][V[j]] != 1:
2974
return False
2975
return True
2976
2977
def has_c4(g):
2978
"""
2979
Tests whether graph ``g`` contains Cycle_4 as an *induced* subgraph.
2980
2981
EXAMPLES:
2982
2983
sage: has_c4(graphs.CycleGraph(4))
2984
True
2985
2986
sage: has_c4(graphs.HouseGraph())
2987
True
2988
2989
sage: has_c4(graphs.CycleGraph(5))
2990
False
2991
2992
sage: has_c4(graphs.DiamondGraph())
2993
False
2994
"""
2995
return g.subgraph_search(c4, induced=True) is not None
2996
2997
def is_c4_free(g):
2998
"""
2999
Tests whether graph ``g`` does not contain Cycle_4 as an *induced* subgraph.
3000
3001
EXAMPLES:
3002
3003
sage: is_c4_free(graphs.CycleGraph(4))
3004
False
3005
3006
sage: is_c4_free(graphs.HouseGraph())
3007
False
3008
3009
sage: is_c4_free(graphs.CycleGraph(5))
3010
True
3011
3012
sage: is_c4_free(graphs.DiamondGraph())
3013
True
3014
"""
3015
return not has_c4(g)
3016
3017
def has_paw(g):
3018
"""
3019
Tests whether graph ``g`` contains a Paw as an *induced* subgraph.
3020
3021
OUTPUT:
3022
3023
Define a Paw to be a 4-vertex graph formed by a triangle and a pendant.
3024
Returns ``True`` if ``g`` contains a Paw as an induced subgraph.
3025
Returns ``False`` otherwise.
3026
3027
EXAMPLES:
3028
3029
sage: has_paw(paw)
3030
True
3031
3032
sage: has_paw(graphs.BullGraph())
3033
True
3034
3035
sage: has_paw(graphs.ClawGraph())
3036
False
3037
3038
sage: has_paw(graphs.DiamondGraph())
3039
False
3040
"""
3041
return g.subgraph_search(paw, induced=True) is not None
3042
3043
def is_paw_free(g):
3044
"""
3045
Tests whether graph ``g`` does not contain a Paw as an *induced* subgraph.
3046
3047
OUTPUT:
3048
3049
Define a Paw to be a 4-vertex graph formed by a triangle and a pendant.
3050
Returns ``False`` if ``g`` contains a Paw as an induced subgraph.
3051
Returns ``True`` otherwise.
3052
3053
EXAMPLES:
3054
3055
sage: is_paw_free(paw)
3056
False
3057
3058
sage: is_paw_free(graphs.BullGraph())
3059
False
3060
3061
sage: is_paw_free(graphs.ClawGraph())
3062
True
3063
3064
sage: is_paw_free(graphs.DiamondGraph())
3065
True
3066
"""
3067
return not has_paw(g)
3068
3069
def has_dart(g):
3070
"""
3071
Tests whether graph ``g`` contains a Dart as an *induced* subgraph.
3072
3073
OUTPUT:
3074
3075
Define a Dart to be a 5-vertex graph formed by ``graphs.DiamondGraph()``
3076
with and a pendant added to one of the degree-3 vertices.
3077
Returns ``True`` if ``g`` contains a Dart as an induced subgraph.
3078
Returns ``False`` otherwise.
3079
3080
EXAMPLES:
3081
3082
sage: has_dart(dart)
3083
True
3084
3085
sage: has_dart(umbrella_4)
3086
True
3087
3088
sage: has_dart(graphs.DiamondGraph())
3089
False
3090
3091
sage: has_dart(bridge)
3092
False
3093
"""
3094
return g.subgraph_search(dart, induced=True) is not None
3095
3096
def is_dart_free(g):
3097
"""
3098
Tests whether graph ``g`` does not contain a Dart as an *induced* subgraph.
3099
3100
OUTPUT:
3101
3102
Define a Dart to be a 5-vertex graph formed by ``graphs.DiamondGraph()``
3103
with and a pendant added to one of the degree-3 vertices.
3104
Returns ``False`` if ``g`` contains a Dart as an induced subgraph.
3105
Returns ``True`` otherwise.
3106
3107
EXAMPLES:
3108
3109
sage: is_dart_free(dart)
3110
False
3111
3112
sage: is_dart_free(umbrella_4)
3113
False
3114
3115
sage: is_dart_free(graphs.DiamondGraph())
3116
True
3117
3118
sage: is_dart_free(bridge)
3119
True
3120
"""
3121
return not has_dart(g)
3122
3123
def is_p4_free(g):
3124
"""
3125
Equivalent to is a cograph - https://en.wikipedia.org/wiki/Cograph
3126
"""
3127
return not has_p4(g)
3128
3129
def has_p4(g):
3130
"""
3131
Tests whether graph ``g`` contains a Path_4 as an *induced* subgraph.
3132
3133
Might also be known as "is not a cograph".
3134
3135
EXAMPLES:
3136
3137
sage: has_p4(graphs.PathGraph(4))
3138
True
3139
3140
sage: has_p4(graphs.CycleGraph(5))
3141
True
3142
3143
sage: has_p4(graphs.CycleGraph(4))
3144
False
3145
3146
sage: has_p4(graphs.CompleteGraph(5))
3147
False
3148
"""
3149
return g.subgraph_search(p4, induced=True) is not None
3150
3151
def has_kite(g):
3152
"""
3153
Tests whether graph ``g`` contains a Kite as an *induced* subgraph.
3154
3155
A Kite is a 5-vertex graph formed by a ``graphs.DiamondGraph()`` with a
3156
pendant attached to one of the degree-2 vertices.
3157
3158
EXAMPLES:
3159
3160
sage: has_kite(kite_with_tail)
3161
True
3162
3163
sage: has_kite(graphs.KrackhardtKiteGraph())
3164
True
3165
3166
sage: has_kite(graphs.DiamondGraph())
3167
False
3168
3169
sage: has_kite(bridge)
3170
False
3171
"""
3172
return g.subgraph_search(kite_with_tail, induced=True) is not None
3173
3174
def is_kite_free(g):
3175
"""
3176
Tests whether graph ``g`` does not contain a Kite as an *induced* subgraph.
3177
3178
A Kite is a 5-vertex graph formed by a ``graphs.DiamondGraph()`` with a
3179
pendant attached to one of the degree-2 vertices.
3180
3181
EXAMPLES:
3182
3183
sage: is_kite_free(kite_with_tail)
3184
False
3185
3186
sage: is_kite_free(graphs.KrackhardtKiteGraph())
3187
False
3188
3189
sage: is_kite_free(graphs.DiamondGraph())
3190
True
3191
3192
sage: is_kite_free(bridge)
3193
True
3194
"""
3195
return not has_kite(g)
3196
3197
def has_claw(g):
3198
"""
3199
Tests whether graph ``g`` contains a Claw as an *induced* subgraph.
3200
3201
A Claw is a 4-vertex graph with one central vertex and 3 pendants.
3202
This is encoded as ``graphs.ClawGraph()``.
3203
3204
EXAMPLES:
3205
3206
sage: has_claw(graphs.ClawGraph())
3207
True
3208
3209
sage: has_claw(graphs.PetersenGraph())
3210
True
3211
3212
sage: has_claw(graphs.BullGraph())
3213
False
3214
3215
sage: has_claw(graphs.HouseGraph())
3216
False
3217
"""
3218
return g.subgraph_search(graphs.ClawGraph(), induced=True) is not None
3219
3220
def is_claw_free(g):
3221
"""
3222
Tests whether graph ``g`` does not contain a Claw as an *induced* subgraph.
3223
3224
A Claw is a 4-vertex graph with one central vertex and 3 pendants.
3225
This is encoded as ``graphs.ClawGraph()``.
3226
3227
EXAMPLES:
3228
3229
sage: is_claw_free(graphs.ClawGraph())
3230
False
3231
3232
sage: is_claw_free(graphs.PetersenGraph())
3233
False
3234
3235
sage: is_claw_free(graphs.BullGraph())
3236
True
3237
3238
sage: is_claw_free(graphs.HouseGraph())
3239
True
3240
"""
3241
return not has_claw(g)
3242
3243
def has_H(g):
3244
"""
3245
Tests whether graph ``g`` contains an H graph as an *induced* subgraph.
3246
3247
An H graph may also be known as a double fork. It is a 6-vertex graph
3248
formed by two Path_3s with their midpoints joined by a bridge.
3249
3250
EXAMPLES:
3251
3252
sage: has_H(double_fork)
3253
True
3254
3255
sage: has_H(graphs.PetersenGraph())
3256
True
3257
3258
sage: has_H(ce71) # double_fork with extra edge
3259
False
3260
3261
sage: has_H(graphs.BullGraph())
3262
False
3263
"""
3264
return g.subgraph_search(double_fork, induced=True) is not None
3265
3266
def is_H_free(g):
3267
"""
3268
Tests if graph ``g`` does not contain a H graph as an *induced* subgraph.
3269
3270
An H graph may also be known as a double fork. It is a 6-vertex graph
3271
formed by two Path_3s with their midpoints joined by a bridge.
3272
3273
EXAMPLES:
3274
3275
sage: is_H_free(double_fork)
3276
False
3277
3278
sage: is_H_free(graphs.PetersenGraph())
3279
False
3280
3281
sage: is_H_free(ce71) # double_fork with extra edge
3282
True
3283
3284
sage: is_H_free(graphs.BullGraph())
3285
True
3286
"""
3287
return not has_H(g)
3288
3289
def has_fork(g):
3290
"""
3291
Tests if graph ``g`` contains a Fork graph as an *induced* subgraph.
3292
3293
A Fork graph may also be known as a Star_1_1_3. It is a 6-vertex graph
3294
formed by a Path_4 with two pendants connected to one end.
3295
It is stored as `star_1_1_3`.
3296
3297
EXAMPLES:
3298
3299
sage: has_fork(star_1_1_3)
3300
True
3301
3302
sage: has_fork(graphs.PetersenGraph())
3303
True
3304
3305
sage: has_fork(graphs.LollipopGraph(3, 2))
3306
False
3307
3308
sage: has_fork(graphs.HouseGraph())
3309
False
3310
3311
sage: has_fork(graphs.ClawGraph())
3312
False
3313
"""
3314
return g.subgraph_search(star_1_1_3, induced=True) is not None
3315
3316
def is_fork_free(g):
3317
"""
3318
Tests if graph ``g`` does not contain Fork graph as an *induced* subgraph.
3319
3320
A Fork graph may also be known as a Star_1_1_3. It is a 6-vertex graph
3321
formed by a Path_4 with two pendants connected to one end.
3322
It is stored as `star_1_1_3`.
3323
3324
EXAMPLES:
3325
3326
sage: is_fork_free(star_1_1_3)
3327
False
3328
3329
sage: is_fork_free(graphs.PetersenGraph())
3330
False
3331
3332
sage: is_fork_free(graphs.LollipopGraph(3, 2))
3333
True
3334
3335
sage: is_fork_free(graphs.HouseGraph())
3336
True
3337
3338
sage: is_fork_free(graphs.ClawGraph())
3339
True
3340
"""
3341
return not has_fork(g)
3342
3343
def has_k4(g):
3344
"""
3345
Tests if graph ``g`` contains a `K_4` as an *induced* subgraph.
3346
3347
`K_4` is the complete graph on 4 vertices.
3348
3349
EXAMPLES:
3350
3351
sage: has_k4(graphs.CompleteGraph(4))
3352
True
3353
3354
sage: has_k4(graphs.CompleteGraph(5))
3355
True
3356
3357
sage: has_k4(graphs.CompleteGraph(3))
3358
False
3359
3360
sage: has_k4(graphs.PetersenGraph())
3361
False
3362
"""
3363
return g.subgraph_search(alpha_critical_easy[2], induced=True) is not None
3364
3365
def is_k4_free(g):
3366
"""
3367
Tests if graph ``g`` does not contain a `K_4` as an *induced* subgraph.
3368
3369
`K_4` is the complete graph on 4 vertices.
3370
3371
EXAMPLES:
3372
3373
sage: is_k4_free(graphs.CompleteGraph(4))
3374
False
3375
3376
sage: is_k4_free(graphs.CompleteGraph(5))
3377
False
3378
3379
sage: is_k4_free(graphs.CompleteGraph(3))
3380
True
3381
3382
sage: is_k4_free(graphs.PetersenGraph())
3383
True
3384
"""
3385
return not has_k4(g)
3386
3387
def is_double_clique(g):
3388
"""
3389
Tests if graph ``g`` can be partitioned into 2 sets which induce cliques.
3390
3391
EXAMPLE:
3392
3393
sage: is_double_clique(p4)
3394
True
3395
3396
sage: is_double_clique(graphs.ButterflyGraph())
3397
True
3398
3399
sage: is_double_clique(graphs.CompleteBipartiteGraph(3,4))
3400
False
3401
3402
sage: is_double_clique(graphs.ClawGraph())
3403
False
3404
3405
sage: is_double_clique(Graph(3))
3406
False
3407
3408
Edge cases ::
3409
3410
sage: is_double_clique(Graph(0))
3411
True
3412
3413
sage: is_double_clique(Graph(1))
3414
True
3415
3416
sage: is_double_clique(Graph(2))
3417
True
3418
"""
3419
gc = g.complement()
3420
return gc.is_bipartite()
3421
3422
def has_radius_equal_diameter(g):
3423
"""
3424
Evaluates whether the radius of graph ``g`` equals its diameter.
3425
3426
Recall the radius of a graph is the minimum eccentricity over all vertices,
3427
or the minimum over all longest distances from a vertex to any other vertex.
3428
Diameter is the maximum eccentricity over all vertices.
3429
Both radius and diamter are defined to be `+Infinity` for disconnected
3430
graphs.
3431
3432
Both radius and diameter are undefined for the empty graph.
3433
3434
EXAMPLES:
3435
3436
sage: has_radius_equal_diameter(Graph(4))
3437
True
3438
3439
sage: has_radius_equal_diameter(graphs.HouseGraph())
3440
True
3441
3442
sage: has_radius_equal_diameter(Graph(1))
3443
True
3444
3445
sage: has_radius_equal_diameter(graphs.ClawGraph())
3446
False
3447
3448
sage: has_radius_equal_diameter(graphs.BullGraph())
3449
False
3450
"""
3451
return g.radius() == g.diameter()
3452
3453
def has_residue_equals_alpha(g):
3454
"""
3455
Evaluate whether the residue of graph ``g`` equals its independence number.
3456
3457
The independence number is the cardinality of the largest independent set
3458
of vertices in ``g``.
3459
The residue of a graph ``g`` with degrees `d_1 \geq d_2 \geq ... \geq d_n`
3460
is found iteratively. First, remove `d_1` from consideration and subtract
3461
`d_1` from the following `d_1` number of elements. Sort. Repeat this
3462
process for `d_2,d_3, ...` until only 0s remain. The number of elements,
3463
i.e. the number of 0s, is the residue of ``g``.
3464
3465
Residue is undefined on the empty graph.
3466
3467
EXAMPLES:
3468
3469
sage: has_residue_equals_alpha(graphs.HouseGraph())
3470
True
3471
3472
sage: has_residue_equals_alpha(graphs.ClawGraph())
3473
True
3474
3475
sage: has_residue_equals_alpha(graphs.CompleteGraph(4))
3476
True
3477
3478
sage: has_residue_equals_alpha(Graph(1))
3479
True
3480
3481
sage: has_residue_equals_alpha(graphs.PetersenGraph())
3482
False
3483
3484
sage: has_residue_equals_alpha(graphs.PathGraph(5))
3485
False
3486
"""
3487
return residue(g) == independence_number(g)
3488
3489
def is_not_forest(g):
3490
"""
3491
Evaluates if graph ``g`` is not a forest.
3492
3493
A forest is a disjoint union of trees. Equivalently, a forest is any acylic
3494
graph, which may or may not be connected.
3495
3496
EXAMPLES:
3497
sage: is_not_forest(graphs.BalancedTree(2,3))
3498
False
3499
3500
sage: is_not_forest(graphs.BalancedTree(2,3).disjoint_union(graphs.BalancedTree(3,3)))
3501
False
3502
3503
sage: is_not_forest(graphs.CycleGraph(5))
3504
True
3505
3506
sage: is_not_forest(graphs.HouseGraph())
3507
True
3508
3509
Edge cases ::
3510
3511
sage: is_not_forest(Graph(1))
3512
False
3513
3514
sage: is_not_forest(Graph(0))
3515
False
3516
"""
3517
return not g.is_forest()
3518
3519
def has_empty_KE_part(g):
3520
"""
3521
Evaluates whether graph ``g`` has an empty Konig-Egervary subgraph.
3522
3523
A Konig-Egervary graph satisfies
3524
independence number + matching number = order.
3525
By [Lar2011]_, every graph contains a unique induced subgraph which is a
3526
Konig-Egervary graph.
3527
3528
EXAMPLES:
3529
3530
sage: has_empty_KE_part(graphs.PetersenGraph())
3531
True
3532
3533
sage: has_empty_KE_part(graphs.CycleGraph(5))
3534
True
3535
3536
sage: has_empty_KE_part(graphs.CompleteBipartiteGraph(3,4))
3537
False
3538
3539
sage: has_empty_KE_part(graphs.CycleGraph(6))
3540
False
3541
3542
Edge cases ::
3543
3544
sage: has_empty_KE_part(Graph(1))
3545
False
3546
3547
sage: has_empty_KE_part(Graph(0))
3548
True
3549
3550
ALGORITHM:
3551
3552
This function is implemented using the Maximum Critical Independent
3553
Set (MCIS) algorithm of [DL2013]_ and applying a Theorem of [Lar2011]_.
3554
3555
Define that an independent set `I` is a critical independent set if
3556
`|I|−|N(I)| \geq |J|−|N(J)|` for any independent set J. Define that a
3557
maximum critical independent set is a critical independent set of maximum
3558
cardinality.
3559
3560
By a Theorem of [Lar2011]_, for every maximum critical independent set `J`
3561
of `G`, the unique Konig-Egervary inducing set `X` is `X = J \cup N(J)`,
3562
where `N(J)` is the neighborhood of `J`.
3563
Therefore, the ``g`` has an empty Konig-Egervary induced subgraph if and
3564
only if the MCIS `J = \emptyset`.
3565
3566
Next, we describe the MCIS algorithm.
3567
Let `B(G) = K_2 \ times G`, i.e. `B(G)` is the bipartite double cover
3568
of `G`. Let `v' \in B(G)` denote the new "twin" of vertex `v \in G`.
3569
Let `a` be the independence number of `B(G)`.
3570
For each vertex `v` in `B(G)`, calculate
3571
`t := independence number(B(G) - \{v,v'\} - N(\{v,v'\})) + 2`.
3572
If `t = a`, then `v` is in the MCIS.
3573
Since we only care about whether the MCIS is empty, if `t = a`,
3574
we return ``False`` and terminate.
3575
3576
Finally, use the Gallai identities to show matching
3577
3578
Finally, we apply the Konig-Egervary Theorem (1931) that for all bipartite
3579
graphs, matching number = vertex cover number. We substitute this into
3580
one of the Gallai identities, that
3581
independence number + covering number = order,
3582
yielding,
3583
independence number = order - matching number.
3584
Since matching number is efficient to compute, our final algorithm is
3585
in fact efficient.
3586
3587
REFERENCES:
3588
3589
.. [DL2013] \Ermelinda DeLaVina and Craig Larson, "A parallel ALGORITHM
3590
for computing the critical independence number and related
3591
sets". ARS Mathematica Contemporanea 6: 237--245, 2013.
3592
3593
.. [Lar2011] \C.E. Larson, "Critical Independent Sets and an
3594
Independence Decomposition Theorem". European Journal of
3595
Combinatorics 32: 294--300, 2011.
3596
"""
3597
b = bipartite_double_cover(g)
3598
alpha = b.order() - b.matching(value_only=True)
3599
for v in g.vertices():
3600
test = b.copy()
3601
test.delete_vertices(closed_neighborhood(b,[(v,0), (v,1)]))
3602
alpha_test = test.order() - test.matching(value_only=True) + 2
3603
if alpha_test == alpha:
3604
return False
3605
return True
3606
3607
def is_class1(g):
3608
"""
3609
Evaluates whether the chomatic index of graph ``g`` equals its max degree.
3610
3611
Let `D` be the maximum degree. By Vizing's Thoerem, all graphs can be
3612
edge-colored in either `D` or `D+1` colors. The case of `D` colors is
3613
called "class 1".
3614
3615
Max degree is undefined for the empty graph.
3616
3617
EXAMPLES:
3618
3619
sage: is_class1(graphs.CompleteGraph(4))
3620
True
3621
3622
sage: is_class1(graphs.WindmillGraph(4,3))
3623
True
3624
3625
sage: is_class1(Graph(1))
3626
True
3627
3628
sage: is_class1(graphs.CompleteGraph(3))
3629
False
3630
3631
sage: is_class1(graphs.PetersenGraph())
3632
False
3633
"""
3634
return g.chromatic_index() == max(g.degree())
3635
3636
def is_class2(g):
3637
"""
3638
Evaluates whether the chomatic index of graph ``g`` equals max degree + 1.
3639
3640
Let `D` be the maximum degree. By Vizing's Thoerem, all graphs can be
3641
edge-colored in either `D` or `D+1` colors. The case of `D+1` colors is
3642
called "class 2".
3643
3644
Max degree is undefined for the empty graph.
3645
3646
EXAMPLES:
3647
3648
sage: is_class2(graphs.CompleteGraph(4))
3649
False
3650
3651
sage: is_class2(graphs.WindmillGraph(4,3))
3652
False
3653
3654
sage: is_class2(Graph(1))
3655
False
3656
3657
sage: is_class2(graphs.CompleteGraph(3))
3658
True
3659
3660
sage: is_class2(graphs.PetersenGraph())
3661
True
3662
"""
3663
return not(g.chromatic_index() == max(g.degree()))
3664
3665
def is_cubic(g):
3666
"""
3667
Evalutes whether graph ``g`` is cubic, i.e. is 3-regular.
3668
3669
EXAMPLES:
3670
3671
sage: is_cubic(graphs.CompleteGraph(4))
3672
True
3673
3674
sage: is_cubic(graphs.PetersenGraph())
3675
True
3676
3677
sage: is_cubic(graphs.CompleteGraph(3))
3678
False
3679
3680
sage: is_cubic(graphs.HouseGraph())
3681
False
3682
"""
3683
D = g.degree()
3684
return min(D) == 3 and max(D) == 3
3685
3686
def is_anti_tutte(g):
3687
"""
3688
Evalutes if graph ``g`` is connected and indep. number <= diameter + girth.
3689
3690
This property is satisfied by many Hamiltonian graphs, but notably not by
3691
the Tutte graph ``graphs.TutteGraph()``.
3692
3693
Diameter is undefined for the empty graph.
3694
3695
EXAMPLES:
3696
3697
sage: is_anti_tutte(graphs.CompleteBipartiteGraph(4, 5))
3698
True
3699
3700
sage: is_anti_tutte(graphs.PetersenGraph())
3701
True
3702
3703
sage: is_anti_tutte(Graph(1))
3704
3705
sage: is_anti_tutte(graphs.TutteGraph())
3706
False
3707
3708
sage: is_anti_tutte(graphs.TutteCoxeterGraph())
3709
False
3710
"""
3711
if not g.is_connected():
3712
return False
3713
return independence_number(g) <= g.diameter() + g.girth()
3714
3715
def is_anti_tutte2(g):
3716
"""
3717
Tests if graph ``g`` has indep. number <= domination number + radius - 1.
3718
3719
``g`` must also be connected.
3720
This property is satisfied by many Hamiltonian graphs, but notably not by
3721
the Tutte graph ``graphs.TutteGraph()``.
3722
3723
Radius is undefined for the empty graph.
3724
3725
EXAMPLES:
3726
3727
sage: is_anti_tutte2(graphs.CompleteGraph(5))
3728
True
3729
3730
sage: is_anti_tutte2(graphs.PetersenGraph())
3731
True
3732
3733
sage: is_anti_tutte2(graphs.TutteGraph())
3734
False
3735
3736
sage: is_anti_tutte2(graphs.TutteCoxeterGraph())
3737
False
3738
3739
sage: is_anti_tutte2(Graph(1))
3740
False
3741
"""
3742
if not g.is_connected():
3743
return False
3744
return independence_number(g) <= domination_number(g) + g.radius()- 1
3745
3746
def diameter_equals_twice_radius(g):
3747
"""
3748
Evaluates whether the diameter of graph ``g`` is equal to twice its radius.
3749
3750
Diameter and radius are undefined for the empty graph.
3751
3752
EXAMPLES:
3753
3754
sage: has_radius_equal_diameter(graphs.ClawGraph())
3755
True
3756
3757
sage: has_radius_equal_diameter(graphs.KrackhardtKiteGraph())
3758
True
3759
3760
sage: diameter_equals_twice_radius(graphs.HouseGraph())
3761
False
3762
3763
sage: has_radius_equal_diameter(graphs.BullGraph())
3764
False
3765
3766
The radius and diameter of ``Graph(1)`` are both 1. ::
3767
3768
sage: diameter_equals_twice_radius(Graph(1))
3769
True
3770
3771
Disconnected graphs have both diameter and radius equal infinity.
3772
3773
sage: diameter_equals_twice_radius(Graph(4))
3774
True
3775
"""
3776
return g.diameter() == 2*g.radius()
3777
3778
def diameter_equals_two(g):
3779
"""
3780
Evaluates whether the diameter of graph ``g`` equals 2.
3781
3782
Diameter is undefined for the empty graph.
3783
3784
EXAMPLES:
3785
3786
sage: diameter_equals_two(graphs.ClawGraph())
3787
True
3788
3789
sage: diameter_equals_two(graphs.HouseGraph())
3790
True
3791
3792
sage: diameter_equals_two(graphs.KrackhardtKiteGraph())
3793
False
3794
3795
sage: diameter_equals_two(graphs.BullGraph())
3796
False
3797
3798
Disconnected graphs have diameter equals infinity.
3799
3800
sage: diameter_equals_two(Graph(4))
3801
False
3802
"""
3803
return g.diameter() == 2
3804
3805
def has_lovasz_theta_equals_alpha(g):
3806
"""
3807
Tests if the Lovasz number of graph ``g`` equals its independence number.
3808
3809
Examples:
3810
3811
sage: has_lovasz_theta_equals_alpha(graphs.CompleteGraph(12))
3812
True
3813
3814
sage: has_lovasz_theta_equals_alpha(double_fork)
3815
True
3816
3817
sage: has_lovasz_theta_equals_alpha(graphs.PetersenGraph())
3818
True
3819
3820
sage: has_lovasz_theta_equals_alpha(graphs.ClebschGraph())
3821
False
3822
3823
sage: has_lovasz_theta_equals_alpha(graphs.CycleGraph(24))
3824
False
3825
3826
True for all graphs with no edges ::
3827
3828
sage: has_lovasz_theta_equals_alpha(Graph(12))
3829
True
3830
3831
Edge cases ::
3832
3833
sage: has_lovasz_theta_equals_alpha(Graph(0))
3834
True
3835
3836
# Broken. Issue #584
3837
sage: has_lovasz_theta_equals_alpha(Graph(1)) # doctest: +SKIP
3838
True
3839
"""
3840
return g.lovasz_theta() == independence_number(g)
3841
3842
def has_lovasz_theta_equals_cc(g):
3843
"""
3844
Test if the Lovasz number of graph ``g`` equals its clique covering number.
3845
3846
Examples:
3847
3848
sage: has_lovasz_theta_equals_cc(graphs.CompleteGraph(12))
3849
True
3850
3851
sage: has_lovasz_theta_equals_cc(double_fork)
3852
True
3853
3854
sage: has_lovasz_theta_equals_cc(graphs.PetersenGraph())
3855
True
3856
3857
sage: has_lovasz_theta_equals_cc(Graph(12))
3858
True
3859
3860
sage: has_lovasz_theta_equals_cc(graphs.ClebschGraph())
3861
False
3862
3863
has_lovasz_theta_equals_alpha(graphs.BuckyBall())
3864
False
3865
3866
Edge cases ::
3867
3868
sage: has_lovasz_theta_equals_cc(Graph(0))
3869
True
3870
3871
# Broken. Issue #584
3872
sage: has_lovasz_theta_equals_cc(Graph(1)) # doctest: +SKIP
3873
True
3874
"""
3875
return g.lovasz_theta() == clique_covering_number(g)
3876
3877
def is_chvatal_erdos(g):
3878
"""
3879
Evaluates whether graph ``g`` meets a Hamiltonicity condition of [CV1972]_.
3880
3881
OUTPUT:
3882
3883
Returns ``True`` if the independence number of ``g`` is less than or equal
3884
to the vertex connectivity of ``g``.
3885
Returns ``False`` otherwise.
3886
3887
EXAMPLES:
3888
3889
sage: is_chvatal_erdos(graphs.CompleteGraph(5))
3890
True
3891
3892
sage: is_chvatal_erdos(graphs.CycleGraph(5))
3893
True
3894
3895
sage: is_chvatal_erdos(graphs.CompleteGraph(2))
3896
True
3897
3898
sage: is_chvatal_erdos(graphs.PetersenGraph())
3899
False
3900
3901
sage: is_chvatal_erdos(graphs.ClawGraph())
3902
False
3903
3904
sage: is_chvatal_erdos(graphs.DodecahedralGraph())
3905
False
3906
3907
Edge cases ::
3908
3909
sage: is_chvatal_erdos(Graph(1))
3910
False
3911
3912
sage: is_chvatal_erdos(Graph(0))
3913
True
3914
3915
REFERENCES:
3916
3917
Theorem: If a graph ``is_chvatal_erdos``, then it is Hamiltonian.
3918
3919
.. [CV1972] \V. Chvatal and P. Erdos, "A note on hamiltonian cycles".
3920
Discrete Mathematics, 2(2): 111--113, 1972.
3921
"""
3922
return independence_number(g) <= g.vertex_connectivity()
3923
3924
def matching_covered(g):
3925
"""
3926
Skipping because broken. See Issue #585.
3927
"""
3928
g = g.copy()
3929
nu = matching_number(g)
3930
E = g.edges()
3931
for e in E:
3932
g.delete_edge(e)
3933
nu2 = matching_number(g)
3934
if nu != nu2:
3935
return False
3936
g.add_edge(e)
3937
return True
3938
3939
def radius_greater_than_center(g):
3940
"""
3941
Test if connected graph ``g`` has radius greater than num. of center verts.
3942
3943
If ``g`` is not connected, returns ``False``.
3944
Radius is undefined for the empty graph.
3945
3946
EXAMPLES:
3947
3948
sage: radius_greater_than_center(graphs.TutteGraph())
3949
True
3950
3951
sage: radius_greater_than_center(graphs.KrackhardtKiteGraph())
3952
True
3953
3954
sage: radius_greater_than_center(graphs.SousselierGraph())
3955
True
3956
3957
sage: radius_greater_than_center(graphs.PetersenGraph())
3958
False
3959
3960
sage: radius_greater_than_center(graphs.DiamondGraph())
3961
False
3962
3963
sage: radius_greater_than_center(Graph(1))
3964
False
3965
"""
3966
return g.is_connected() and g.radius() > card_center(g)
3967
3968
def avg_distance_greater_than_girth(g):
3969
"""
3970
Tests if graph ``g`` is connected and avg. distance greater than the girth.
3971
3972
Average distance is undefined for 1- and 0- vertex graphs.
3973
3974
EXAMPLES:
3975
3976
sage: avg_distance_greater_than_girth(graphs.TutteGraph())
3977
True
3978
3979
sage: avg_distance_greater_than_girth(graphs.HarborthGraph())
3980
True
3981
3982
sage: avg_distance_greater_than_girth(graphs.HortonGraph())
3983
True
3984
3985
sage: avg_distance_greater_than_girth(graphs.BullGraph())
3986
False
3987
3988
sage: avg_distance_greater_than_girth(Graph("NC`@A?_C?@_JA??___W"))
3989
False
3990
3991
sage: avg_distance_greater_than_girth(Graph(2))
3992
False
3993
3994
Acyclic graphs have girth equals infinity. ::
3995
3996
sage: avg_distance_greater_than_girth(graphs.CompleteGraph(2))
3997
False
3998
"""
3999
return g.is_connected() and g.average_distance() > g.girth()
4000
4001
def chi_equals_min_theory(g):
4002
"""
4003
Evaluate if chromatic num. of graph ``g`` equals min. of some upper bounds.
4004
4005
Some known upper bounds on the chromatic number Chi (`\chi`) include
4006
our invariants `[brooks, wilf, welsh_powell, szekeres_wilf]`.
4007
Returns ``True`` if the actual chromatic number of ``g`` equals the minimum
4008
of / "the best of" these known upper bounds.
4009
4010
Some of these invariants are undefined on the empty graph.
4011
4012
EXAMPLES:
4013
4014
sage: chi_equals_min_theory(Graph(1))
4015
True
4016
4017
sage: chi_equals_min_theory(graphs.PetersenGraph())
4018
True
4019
4020
sage: chi_equals_min_theory(double_fork)
4021
True
4022
4023
sage: chi_equals_min_theory(Graph(3))
4024
False
4025
4026
chi_equals_min_theory(graphs.CompleteBipartiteGraph(3,5))
4027
False
4028
4029
chi_equals_min_theory(graphs.IcosahedralGraph())
4030
False
4031
"""
4032
chromatic_upper_theory = [brooks, wilf, welsh_powell, szekeres_wilf]
4033
min_theory = min([f(g) for f in chromatic_upper_theory])
4034
return min_theory == g.chromatic_number()
4035
4036
def is_heliotropic_plant(g):
4037
"""
4038
Evaluates whether graph ``g`` is a heliotropic plant. BROKEN
4039
4040
BROKEN: code should be nonnegative eigen, not just positive eigen.
4041
See Issue #586
4042
4043
A graph is heliotropic iff the independence number equals the number of
4044
nonnegative eigenvalues.
4045
4046
See [BDF1995]_ for a definition and some related conjectures, where
4047
[BDF1995]_ builds on the conjecturing work of Siemion Fajtlowicz.
4048
4049
EXAMPLES:
4050
4051
REFERENCES:
4052
4053
.. [BDF1995] Tony Brewster, Michael J.Dinneen, and Vance Faber, "A
4054
computational attack on the conjectures of Graffiti: New
4055
counterexamples and proofs". Discrete Mathematics,
4056
147(1--3): 35--55, 1995.
4057
"""
4058
return (independence_number(g) == card_positive_eigenvalues(g))
4059
4060
def is_geotropic_plant(g):
4061
"""
4062
Evaluates whether graph ``g`` is a heliotropic plant. BROKEN
4063
4064
BROKEN: code should be nonpositive eigen, not just negative eigen.
4065
See Issue #586
4066
4067
A graph is geotropic iff the independence number equals the number of
4068
nonnegative eigenvalues.
4069
4070
See [BDF1995]_ for a definition and some related conjectures, where
4071
[BDF1995]_ builds on the conjecturing work of Siemion Fajtlowicz.
4072
4073
EXAMPLES:
4074
4075
REFERENCES:
4076
4077
.. [BDF1995] Tony Brewster, Michael J.Dinneen, and Vance Faber, "A
4078
computational attack on the conjectures of Graffiti: New
4079
counterexamples and proofs". Discrete Mathematics,
4080
147(1--3): 35--55, 1995.
4081
"""
4082
return (independence_number(g) == card_negative_eigenvalues(g))
4083
4084
def is_traceable(g):
4085
"""
4086
Evaluates whether graph ``g`` is traceable.
4087
4088
A graph ``g`` is traceable iff there exists a Hamiltonian path, i.e. a path
4089
which visits all vertices in ``g`` once.
4090
This is different from ``is_hamiltonian``, since that checks if there
4091
exists a Hamiltonian *cycle*, i.e. a path which then connects backs to
4092
its starting point.
4093
4094
EXAMPLES:
4095
4096
sage: is_traceable(graphs.CompleteGraph(5))
4097
True
4098
4099
sage: is_traceable(graphs.PathGraph(5))
4100
True
4101
4102
sage: is_traceable(graphs.PetersenGraph())
4103
True
4104
4105
sage: is_traceable(graphs.CompleteGraphs(2))
4106
True
4107
4108
sage: is_traceable(Graph(3))
4109
False
4110
4111
sage: is_traceable(graphs.ClawGraph())
4112
False
4113
4114
sage: is_traceable(graphs.ButterflyGraph())
4115
False
4116
4117
Edge cases ::
4118
4119
sage: is_traceable(Graph(0))
4120
False
4121
4122
sage: is_traceable(Graph(1))
4123
False
4124
4125
ALGORITHM:
4126
4127
A graph `G` is traceable iff the join `G'` of `G` with a single new vertex
4128
`v` is Hamiltonian, where join means to connect every vertex of `G` to the
4129
new vertex `v`.
4130
Why? Suppose there exists a Hamiltonian path between `u` and `w` in `G`.
4131
Then, in `G'`, make a cycle from `v` to `u` to `w` and back to `v`.
4132
For the reverse direction, just note that the additional vertex `v` cannot
4133
"help" since Hamiltonian paths can only visit any vertex once.
4134
"""
4135
gadd = g.join(Graph(1),labels="integers")
4136
return gadd.is_hamiltonian()
4137
4138
def has_residue_equals_two(g):
4139
"""
4140
Evaluates whether the residue of graph ``g`` equals 2.
4141
4142
The residue of a graph ``g`` with degrees `d_1 \geq d_2 \geq ... \geq d_n`
4143
is found iteratively. First, remove `d_1` from consideration and subtract
4144
`d_1` from the following `d_1` number of elements. Sort. Repeat this
4145
process for `d_2,d_3, ...` until only 0s remain. The number of elements,
4146
i.e. the number of 0s, is the residue of ``g``.
4147
4148
Residue is undefined on the empty graph.
4149
4150
EXAMPLES:
4151
4152
sage: has_residue_equals_two(graphs.ButterflyGraph())
4153
True
4154
4155
sage: has_residue_equals_two(graphs.IcosahedralGraph())
4156
True
4157
4158
sage: has_residue_equals_two(graphs.OctahedralGraph())
4159
True
4160
4161
sage: has_residue_equals_two(Graph(1))
4162
False
4163
4164
sage: has_residue_equals_two(graphs.BullGraph())
4165
False
4166
4167
sage: has_residue_equals_two(graphs.BrinkmannGraph())
4168
False
4169
"""
4170
return residue(g) == 2
4171
4172
def is_chordal_or_not_perfect(g):
4173
"""
4174
Evaluates if graph ``g`` is either chordal or not perfect.
4175
4176
There is a known theorem that every chordal graph is perfect.
4177
4178
OUTPUT:
4179
4180
Returns ``True`` iff ``g`` is chordal OR (inclusive or) ``g`` is not
4181
perfect.
4182
4183
EXAMPLES:
4184
4185
sage: is_chordal_or_not_perfect(graphs.DiamondGraph())
4186
True
4187
4188
sage: is_chordal_or_not_perfect(graphs.CycleGraph(5))
4189
True
4190
4191
sage: is_chordal_or_not_perfect(graphs.LollipopGraph(5,3))
4192
True
4193
4194
sage: is_chordal_or_not_perfect(graphs.CycleGraph(4))
4195
False
4196
4197
sage: is_chordal_or_not_perfect(graphs.HexahedralGraph())
4198
False
4199
4200
Vacuously chordal cases ::
4201
4202
sage: is_chordal_or_not_perfect(Graph(0))
4203
True
4204
4205
sage: is_chordal_or_not_perfect(Graph(1))
4206
True
4207
4208
sage: is_complement_of_chordal(Graph(4))
4209
True
4210
"""
4211
if g.is_chordal():
4212
return true
4213
else:
4214
return not g.is_perfect()
4215
4216
def has_alpha_residue_equal_two(g):
4217
"""
4218
Tests if both the residue and independence number of graphs ``g`` equal 2.
4219
4220
The residue of a graph ``g`` with degrees `d_1 \geq d_2 \geq ... \geq d_n`
4221
is found iteratively. First, remove `d_1` from consideration and subtract
4222
`d_1` from the following `d_1` number of elements. Sort. Repeat this
4223
process for `d_2,d_3, ...` until only 0s remain. The number of elements,
4224
i.e. the number of 0s, is the residue of ``g``.
4225
4226
Residue is undefined on the empty graph.
4227
4228
EXAMPLES:
4229
4230
sage: has_alpha_residue_equal_two(graphs.DiamondGraph())
4231
True
4232
4233
sage: has_alpha_residue_equal_two(Graph(2))
4234
True
4235
4236
sage: has_alpha_residue_equal_two(graphs.OctahedralGraph())
4237
True
4238
4239
sage: has_alpha_residue_equal_two(graphs.BullGraph())
4240
False
4241
4242
sage: has_alpha_residue_equal_two(graphs.BidiakisCube())
4243
False
4244
4245
sage: has_alpha_residue_equal_two(Graph(3))
4246
False
4247
4248
sage: has_alpha_residue_equal_two(Graph(1))
4249
False
4250
"""
4251
if residue(g) != 2:
4252
return false
4253
else:
4254
return independence_number(g) == 2
4255
4256
def alpha_leq_order_over_two(g):
4257
"""
4258
Tests if the independence number of graph ``g`` is at most half its order.
4259
4260
EXAMPLES:
4261
4262
sage: alpha_leq_order_over_two(graphs.ButterflyGraph())
4263
True
4264
4265
sage: alpha_leq_order_over_two(graphs.DiamondGraph())
4266
True
4267
4268
sage: alpha_leq_order_over_two(graphs.CoxeterGraph())
4269
True
4270
4271
sage: alpha_leq_order_over_two(Graph(4))
4272
False
4273
4274
sage: alpha_leq_order_over_two(graphs.BullGraph())
4275
False
4276
4277
Edge cases ::
4278
4279
sage: alpha_leq_order_over_two(Graph(0))
4280
True
4281
4282
sage: alpha_leq_order_over_two(Graph(1))
4283
False
4284
"""
4285
return (2*independence_number(g) <= g.order())
4286
4287
def order_leq_twice_max_degree(g):
4288
"""
4289
Tests if the order of graph ``g`` is at most twice the max of its degrees.
4290
4291
Undefined for the empty graph.
4292
4293
EXAMPLES:
4294
4295
sage: order_leq_twice_max_degree(graphs.BullGraph())
4296
True
4297
4298
sage: order_leq_twice_max_degree(graphs.ThomsenGraph())
4299
True
4300
4301
sage: order_leq_twice_max_degree(graphs.CycleGraph(4))
4302
True
4303
4304
sage: order_leq_twice_max_degree(graphs.BidiakisCube())
4305
False
4306
4307
sage: order_leq_twice_max_degree(graphs.CycleGraph(5))
4308
False
4309
4310
sage: order_leq_twice_max_degree(Graph(1))
4311
False
4312
"""
4313
return (g.order() <= 2*max(g.degree()))
4314
4315
def is_chromatic_index_critical(g):
4316
"""
4317
Evaluates whether graph ``g`` is chromatic index critical.
4318
4319
Let `\chi(G)` denote the chromatic index of a graph `G`.
4320
Then `G` is chromatic index critical if `\chi(G-e) < \chi(G)` (strictly
4321
less than) for all `e \in G` AND if (by definition) `G` is class 2.
4322
4323
See [FW1977]_ for a more extended definition and discussion.
4324
4325
We initially found it surprising that `G` is required to be class 2; for
4326
example, the Star Graph is a class 1 graph which satisfies the rest of
4327
the definition. We have found articles which equivalently define critical
4328
graphs as class 2 graphs which become class 1 when any edge is removed.
4329
Perhaps this latter definition inspired the one we state above?
4330
4331
Max degree is undefined on the empty graph, so ``is_class`` is also
4332
undefined. Therefore this property is undefined on the empty graph.
4333
4334
EXAMPLES:
4335
4336
sage: is_chromatic_index_critical(Graph('Djk'))
4337
True
4338
4339
sage: is_chromatic_index_critical(graphs.CompleteGraph(3))
4340
True
4341
4342
sage: is_chromatic_index_critical(graphs.CycleGraph(5))
4343
True
4344
4345
sage: is_chromatic_index_critical(graphs.CompleteGraph(5))
4346
False
4347
4348
sage: is_chromatic_index_critical(graphs.PetersenGraph())
4349
False
4350
4351
sage: is_chromatic_index_critical(graphs.FlowerSnark())
4352
False
4353
4354
Non-trivially disconnected graphs ::
4355
4356
sage: is_chromatic_index_critical(graphs.CycleGraph(4).disjoint_union(graphs.CompleteGraph(4)))
4357
False
4358
4359
Class 1 graphs ::
4360
4361
sage: is_chromatic_index_critical(Graph(1))
4362
False
4363
4364
sage: is_chromatic_index_critical(graphs.CompleteGraph(4))
4365
False
4366
4367
sage: is_chromatic_index_critical(graphs.CompleteGraph(2))
4368
False
4369
4370
sage: is_chromatic_index_critical(graphs.StarGraph(4))
4371
False
4372
4373
ALGORITHM:
4374
4375
This function uses a series of tricks to reduce the number of cases that
4376
need to be considered, before finally checking in the obvious way.
4377
4378
First, if a graph has more than 1 non-trivial connected component, then
4379
return ``False``. This is because in a graph with multiple such components,
4380
removing any edges from the smaller component cannot affect the chromatic
4381
index.
4382
4383
Second, check if the graph is class 2. If not, stop and return ``False``.
4384
4385
Finally, identify isomorphic edges using the line graph and its orbits.
4386
We then need only check the non-equivalent edges to see that they reduce
4387
the chromatic index when deleted.
4388
4389
REFERENCES:
4390
4391
.. [FW1977] \S. Fiorini and R.J. Wilson, "Edge-colourings of graphs".
4392
Pitman Publishing, London, UK, 1977.
4393
"""
4394
component_sizes = g.connected_components_sizes()
4395
if len(component_sizes) > 1:
4396
if component_sizes[1] > 1:
4397
return False
4398
4399
if chi == max_degree(g):
4400
return False
4401
4402
lg = g.line_graph()
4403
equiv_lines = lg.automorphism_group(return_group=False, orbits=true)
4404
equiv_lines_representatives = [orb[0] for orb in equiv_lines]
4405
4406
gc = g.copy()
4407
for e in equiv_lines_representatives:
4408
gc.delete_edge(e)
4409
chi_prime = gc.chromatic_index()
4410
if chi_prime == chi:
4411
return False
4412
gc.add_edge(e)
4413
return True
4414
4415
#alpha(g-e) > alpha(g) for *every* edge g
4416
def is_alpha_critical(g):
4417
#if not g.is_connected():
4418
#return False
4419
alpha = independence_number(g)
4420
for e in g.edges():
4421
gc = copy(g)
4422
gc.delete_edge(e)
4423
alpha_prime = independence_number(gc)
4424
if alpha_prime <= alpha:
4425
return False
4426
return True
4427
4428
#graph is KE if matching number + independence number = n, test does *not* compute alpha
4429
def is_KE(g):
4430
return g.order() == len(find_KE_part(g))
4431
4432
#graph is KE if matching number + independence number = n, test comoutes alpha
4433
#def is_KE(g):
4434
# return (g.matching(value_only = True) + independence_number(g) == g.order())
4435
4436
#possibly faster version of is_KE (not currently in invariants)
4437
#def is_KE2(g):
4438
# return (independence_number(g) == critical_independence_number(g))
4439
4440
def is_independence_irreducible(g):
4441
return g.order() == card_independence_irreducible_part(g)
4442
4443
4444
def is_factor_critical(g):
4445
"""
4446
a graph is factor-critical if order is odd and removal of any vertex gives graph with perfect matching
4447
is_factor_critical(graphs.PathGraph(3))
4448
False
4449
sage: is_factor_critical(graphs.CycleGraph(5))
4450
True
4451
"""
4452
if g.order() % 2 == 0:
4453
return False
4454
for v in g.vertices():
4455
gc = copy(g)
4456
gc.delete_vertex(v)
4457
if not gc.has_perfect_matching:
4458
return False
4459
return True
4460
4461
#returns a list of (necessarily non-adjacent) vertices that have the same neighbors as v if a pair exists or None
4462
def find_twins_of_vertex(g,v):
4463
L = []
4464
V = g.vertices()
4465
D = g.distance_all_pairs()
4466
for i in range(g.order()):
4467
w = V[i]
4468
if D[v][w] == 2 and g.neighbors(v) == g.neighbors(w):
4469
L.append(w)
4470
return L
4471
4472
def has_twin(g):
4473
t = find_twin(g)
4474
if t == None:
4475
return False
4476
else:
4477
return True
4478
4479
def is_twin_free(g):
4480
return not has_twin(g)
4481
4482
#returns twin pair (v,w) if one exists, else returns None
4483
#can't be adjacent
4484
def find_twin(g):
4485
4486
V = g.vertices()
4487
for v in V:
4488
Nv = set(g.neighbors(v))
4489
for w in V:
4490
Nw = set(g.neighbors(w))
4491
if v not in Nw and Nv == Nw:
4492
return (v,w)
4493
return None
4494
4495
def find_neighbor_twins(g):
4496
V = g.vertices()
4497
for v in V:
4498
Nv = g.neighbors(v)
4499
for w in Nv:
4500
if set(closed_neighborhood(g,v)) == set(closed_neighborhood(g,w)):
4501
return (v,w)
4502
return None
4503
4504
#given graph g and subset S, looks for any neighbor twin of any vertex in T
4505
#if result = T, then no twins, else the result is maximal, but not necessarily unique
4506
def find_neighbor_twin(g, T):
4507
gT = g.subgraph(T)
4508
for v in T:
4509
condition = False
4510
Nv = set(g.neighbors(v))
4511
#print "v = {}, Nv = {}".format(v,Nv)
4512
NvT = set(gT.neighbors(v))
4513
for w in Nv:
4514
NwT = set(g.neighbors(w)).intersection(set(T))
4515
if w not in T and NvT.issubset(NwT):
4516
T.append(w)
4517
condition = True
4518
#print "TWINS: v = {}, w = {}, sp3 = {}".format(v,w,sp3)
4519
break
4520
if condition == True:
4521
break
4522
4523
#if result = T, then no twins, else the result is maximal, but not necessarily unique
4524
def iterative_neighbor_twins(g, T):
4525
T2 = copy(T)
4526
find_neighbor_twin(g, T)
4527
while T2 != T:
4528
T2 = copy(T)
4529
find_neighbor_twin(g, T)
4530
return T
4531
4532
4533
#can't compute membership in this class directly. instead testing isomorhism for 400 known class0 graphs
4534
def is_pebbling_class0(g):
4535
for hkey in class0graphs_dict:
4536
h = Graph(class0graphs_dict[hkey])
4537
if g.is_isomorphic(h):
4538
return True
4539
return False
4540
4541
def girth_greater_than_2log(g):
4542
return bool(g.girth() > 2*log(g.order(),2))
4543
4544
def szekeres_wilf_equals_chromatic_number(g):
4545
return szekeres_wilf(g) == g.chromatic_number()
4546
4547
def has_Havel_Hakimi_property(g, v):
4548
"""
4549
This function returns whether the vertex v in the graph g has the Havel-Hakimi
4550
property as defined in [1]. A vertex has the Havel-Hakimi property if it has
4551
maximum degree and the minimum degree of its neighbours is at least the maximum
4552
degree of its non-neigbours.
4553
4554
[1] Graphs with the strong Havel-Hakimi property, M. Barrus, G. Molnar, Graphs
4555
and Combinatorics, 2016, http://dx.doi.org/10.1007/s00373-015-1674-7
4556
4557
Every vertex in a regular graph has the Havel-Hakimi property::
4558
4559
sage: P = graphs.PetersenGraph()
4560
sage: for v in range(10):
4561
....: has_Havel_Hakimi_property(P,v)
4562
True
4563
True
4564
True
4565
True
4566
True
4567
True
4568
True
4569
True
4570
True
4571
True
4572
sage: has_Havel_Hakimi_property(Graph([[0,1,2,3],lambda x,y: False]),0)
4573
True
4574
sage: has_Havel_Hakimi_property(graphs.CompleteGraph(5),0)
4575
True
4576
"""
4577
if max_degree(g) > g.degree(v): return False
4578
4579
#handle the case where the graph is an independent set
4580
if len(g.neighbors(v)) == 0: return True
4581
4582
#handle the case where v is adjacent with all vertices
4583
if len(g.neighbors(v)) == len(g.vertices()) - 1: return True
4584
4585
return (min(g.degree(nv) for nv in g.neighbors(v)) >=
4586
max(g.degree(nnv) for nnv in g.vertices() if nnv != v and nnv not in g.neighbors(v)))
4587
4588
def has_strong_Havel_Hakimi_property(g):
4589
"""
4590
This function returns whether the graph g has the strong Havel-Hakimi property
4591
as defined in [1]. A graph has the strong Havel-Hakimi property if in every
4592
induced subgraph H of G, every vertex of maximum degree has the Havel-Hakimi
4593
property.
4594
4595
[1] Graphs with the strong Havel-Hakimi property, M. Barrus, G. Molnar, Graphs
4596
and Combinatorics, 2016, http://dx.doi.org/10.1007/s00373-015-1674-7
4597
4598
The graph obtained by connecting two cycles of length 3 by a single edge has
4599
the strong Havel-Hakimi property::
4600
4601
sage: has_strong_Havel_Hakimi_property(Graph('E{CW'))
4602
True
4603
"""
4604
for S in Subsets(g.vertices()):
4605
if len(S)>2:
4606
H = g.subgraph(S)
4607
Delta = max_degree(H)
4608
if any(not has_Havel_Hakimi_property(H, v) for v in S if H.degree(v) == Delta):
4609
return False
4610
return True
4611
4612
# Graph is subcubic is each vertex is at most degree 3
4613
def is_subcubic(g):
4614
return max_degree(g) <= 3
4615
4616
# Max and min degree varies by at most 1
4617
def is_quasi_regular(g):
4618
if max_degree(g) - min_degree(g) < 2:
4619
return true
4620
return false
4621
4622
# g is bad if a block is isomorphic to k3, c5, k4*, c5*
4623
def is_bad(g):
4624
blocks = g.blocks_and_cut_vertices()[0]
4625
# To make a subgraph of g from the ith block
4626
for i in blocks:
4627
h = g.subgraph(i)
4628
boolean = h.is_isomorphic(alpha_critical_easy[1]) or h.is_isomorphic(alpha_critical_easy[4]) or h.is_isomorphic(alpha_critical_easy[5]) or h.is_isomorphic(alpha_critical_easy[21])
4629
if boolean == True:
4630
return True
4631
return False
4632
4633
# Graph g is complement_hamiltonian if the complement of the graph is hamiltonian.
4634
def is_complement_hamiltonian(g):
4635
return g.complement().is_hamiltonian()
4636
4637
# A graph is unicyclic if it is connected and has order == size
4638
# Equivalently, graph is connected and has exactly one cycle
4639
def is_unicyclic(g):
4640
"""
4641
Tests:
4642
sage: is_unicyclic(graphs.BullGraph())
4643
True
4644
sage: is_unicyclic(graphs.ButterflyGraph())
4645
False
4646
"""
4647
return g.is_connected() and g.order() == g.size()
4648
4649
def is_k_tough(g,k):
4650
return toughness(g) >= k # In invariants
4651
def is_1_tough(g):
4652
return is_k_tough(g, 1)
4653
def is_2_tough(g):
4654
return is_k_tough(g, 2)
4655
4656
# True if graph has at least two hamiltonian cycles. The cycles may share some edges.
4657
def has_two_ham_cycles(gIn):
4658
g = gIn.copy()
4659
g.relabel()
4660
try:
4661
ham1 = g.hamiltonian_cycle()
4662
except EmptySetError:
4663
return False
4664
4665
for e in ham1.edges():
4666
h = copy(g)
4667
h.delete_edge(e)
4668
if h.is_hamiltonian():
4669
return true
4670
return false
4671
4672
def has_simplical_vertex(g):
4673
"""
4674
v is a simplical vertex if induced neighborhood is a clique.
4675
"""
4676
for v in g.vertices():
4677
if is_simplical_vertex(g, v):
4678
return true
4679
return false
4680
4681
def has_exactly_two_simplical_vertices(g):
4682
"""
4683
v is a simplical vertex if induced neighborhood is a clique.
4684
"""
4685
return simplical_vertices(g) == 2
4686
4687
def is_two_tree(g):
4688
"""
4689
Define k-tree recursively:
4690
- Complete Graph on (k+1)-vertices is a k-tree
4691
- A k-tree on n+1 vertices is constructed by selecting some k-tree on n vertices and
4692
adding a degree k vertex such that its open neighborhood is a clique.
4693
"""
4694
if(g.is_isomorphic(graphs.CompleteGraph(3))):
4695
return True
4696
4697
# We can just recurse from any degree-2 vertex; no need to test is_two_tree(g-w) if is_two_tree(g-v) returns False.
4698
# Intuition why: if neighborhood of a degree-2 v is not a triangle, it won't become one if we remove w (so clique check OK),
4699
# and, removing a degree-2 vertex of one triangle cannot destroy another triangle (so recursion OK).
4700
degree_two_vertices = (v for v in g.vertices() if g.degree(v) == 2)
4701
try:
4702
v = next(degree_two_vertices)
4703
except StopIteration: # Empty list. No degree 2 vertices.
4704
return False
4705
4706
if not g.has_edge(g.neighbors(v)): # Clique
4707
return false
4708
g2 = g.copy()
4709
g2.delete_vertex(v)
4710
return is_two_tree(g2)
4711
4712
def is_two_path(g):
4713
"""
4714
Graph g is a two_path if it is a two_tree and has exactly 2 simplical vertices
4715
"""
4716
return has_exactly_two_simplical_vertices(g) and is_two_tree(g)
4717
4718
def is_prism_hamiltonian(g):
4719
"""
4720
A graph G is prism hamiltonian if G x K2 (cartesian product) is hamiltonian
4721
"""
4722
return g.cartesian_product(graphs.CompleteGraph(2)).is_hamiltonian()
4723
4724
# Bauer, Douglas, et al. "Long cycles in graphs with large degree sums." Discrete Mathematics 79.1 (1990): 59-70.
4725
def is_bauer(g):
4726
"""
4727
True if g is 2_tough and sigma_3 >= order
4728
"""
4729
return is_2_tough(g) and sigma_k(g, 3) >= g.order()
4730
4731
# Jung, H. A. "On maximal circuits in finite graphs." Annals of Discrete Mathematics. Vol. 3. Elsevier, 1978. 129-144.
4732
def is_jung(g):
4733
"""
4734
True if graph has n >= 11, if graph is 1-tough, and sigma_2 >= n - 4.
4735
See functions toughness(g) and sigma_2(g) for more details.
4736
"""
4737
return g.order() >= 11 and is_1_tough(g) and sigma_2(g) >= g.order() - 4
4738
4739
# Bela Bollobas and Andrew Thomason, Weakly Pancyclic Graphs. Journal of Combinatorial Theory 77: 121--137, 1999.
4740
def is_weakly_pancyclic(g):
4741
"""
4742
True if g contains cycles of every length k from k = girth to k = circumfrence
4743
4744
Returns False if g is acyclic (in which case girth = circumfrence = +Infinity).
4745
4746
sage: is_weakly_pancyclic(graphs.CompleteGraph(6))
4747
True
4748
sage: is_weakly_pancyclic(graphs.PetersenGraph())
4749
False
4750
"""
4751
lengths = cycle_lengths(g)
4752
if not lengths: # acyclic
4753
raise ValueError("Graph is acyclic. Property undefined.")
4754
else:
4755
return lengths == set(range(min(lengths),max(lengths)+1))
4756
4757
def is_pancyclic(g):
4758
"""
4759
True if g contains cycles of every length from 3 to g.order()
4760
4761
sage: is_pancyclic(graphs.OctahedralGraph())
4762
True
4763
sage: is_pancyclic(graphs.CycleGraph(10))
4764
False
4765
"""
4766
lengths = cycle_lengths(g)
4767
return lengths == set(range(3, g.order()+1))
4768
4769
def has_two_walk(g):
4770
"""
4771
A two-walk is a closed walk that visits every vertex and visits no vertex more than twice.
4772
4773
Two-walk is a generalization of Hamiltonian cycles. If a graph is Hamiltonian, then it has a two-walk.
4774
4775
sage: has_two_walk(c4c4)
4776
True
4777
sage: has_two_walk(graphs.WindmillGraph(3,3))
4778
False
4779
"""
4780
for init_vertex in g.vertices():
4781
path_stack = [[init_vertex]]
4782
while path_stack:
4783
path = path_stack.pop()
4784
for neighbor in g.neighbors(path[-1]):
4785
if neighbor == path[0] and all(v in path for v in g.vertices()):
4786
return True
4787
elif path.count(neighbor) < 2:
4788
path_stack.append(path + [neighbor])
4789
return False
4790
4791
def is_claw_free_paw_free(g):
4792
return is_claw_free(g) and is_paw_free(g)
4793
4794
def has_bull(g):
4795
"""
4796
True if g has an induced subgraph isomorphic to graphs.BullGraph()
4797
"""
4798
return g.subgraph_search(graphs.BullGraph(), induced = True) != None
4799
4800
def is_bull_free(g):
4801
"""
4802
True if g does not have an induced subgraph isomoprhic to graphs.BullGraph()
4803
"""
4804
return not has_bull(g)
4805
4806
def is_claw_free_bull_free(g):
4807
return is_claw_free(g) and is_bull_free(g)
4808
4809
def has_F(g):
4810
"""
4811
Let F be a triangle with 3 pendants. True if g has an induced F.
4812
"""
4813
F = graphs.CycleGraph(3)
4814
F.add_vertices([3,4,5])
4815
F.add_edges([(0,3), [1,4], [2,5]])
4816
return g.subgraph_search(F, induced = True) != None
4817
4818
def is_F_free(g):
4819
"""
4820
Let F be a triangle with 3 pendants. True if g has no induced F.
4821
"""
4822
return not has_F(g)
4823
4824
# Ronald Gould, Updating the Hamiltonian problem — a survey. Journal of Graph Theory 15.2: 121-157, 1991.
4825
def is_oberly_sumner(g):
4826
"""
4827
g is_oberly_sumner if order >= 3, is_two_connected, is_claw_free, AND is_F_free
4828
"""
4829
return g.order() >= 3 and is_two_connected(g) and is_claw_free(g) and is_F_free(g)
4830
def is_oberly_sumner_bull(g):
4831
"""
4832
True if g is 2-connected, claw-free, and bull-free
4833
"""
4834
return is_two_connected(g) and is_claw_free_bull_free(g)
4835
def is_oberly_sumner_p4(g):
4836
"""
4837
True if g is 2-connected, claw-free, and p4-free
4838
"""
4839
return is_two_connected(g) and is_claw_free(g) and is_p4_free(g)
4840
4841
# Ronald Gould, Updating the Hamiltonian problem — a survey. Journal of Graph Theory 15.2: 121-157, 1991.
4842
def is_matthews_sumner(g):
4843
"""
4844
True if g is 2-connected, claw-free, and minimum-degree >= (order-1) / 3
4845
"""
4846
return is_two_connected(g) and is_claw_free(g) and min_degree(g) >= (g.order() - 1) / 3
4847
def is_broersma_veldman_gould(g):
4848
"""
4849
True if g is 2-connected, claw-free, and diameter <= 2
4850
"""
4851
return is_two_connected(g) and is_claw_free(g) and g.diameter() <= 2
4852
4853
def chvatals_condition(g):
4854
"""
4855
True if g.order()>=3 and given increasing degrees d_1,..,d_n, for all i, i>=n/2 or d_i>i or d_{n-i}>=n-1
4856
4857
This condition is based on Thm 1 of
4858
Chvátal, Václav. "On Hamilton's ideals." Journal of Combinatorial Theory, Series B 12.2 (1972): 163-168.
4859
4860
[Chvatal, 72] also showed this condition is sufficient to imply g is Hamiltonian.
4861
"""
4862
if g.order() < 3:
4863
return False
4864
degrees = g.degree()
4865
degrees.sort()
4866
n = g.order()
4867
return all(degrees[i] > i or i >= n/2 or degrees[n-i] >= n-i for i in range(0, len(degrees)))
4868
4869
def is_matching(g):
4870
"""
4871
Returns True if this graph is the disjoint union of complete graphs of order 2.
4872
4873
Tests:
4874
sage: is_matching(graphs.CompleteGraph(2))
4875
True
4876
sage: is_matching(graphs.PathGraph(4))
4877
False
4878
sage: is_matching(graphs.CompleteGraph(2).disjoint_union(graphs.CompleteGraph(2)))
4879
True
4880
"""
4881
return min(g.degree())==1 and max(g.degree())==1
4882
4883
def has_odd_order(g):
4884
"""
4885
True if the number of vertices in g is odd
4886
4887
sage: has_odd_order(Graph(5))
4888
True
4889
sage: has_odd_order(Graph(2))
4890
False
4891
"""
4892
return g.order() % 2 == 1
4893
4894
def has_even_order(g):
4895
"""
4896
True if the number of vertices in g is even
4897
4898
sage: has_even_order(Graph(5))
4899
False
4900
sage: has_even_order(Graph(2))
4901
True
4902
"""
4903
return g.order() % 2 == 0
4904
4905
def is_maximal_triangle_free(g):
4906
"""
4907
Evaluates whether graphs ``g`` is a maximal triangle-free graph
4908
4909
Maximal triangle-free means that adding any edge to ``g`` will create a
4910
triangle.
4911
If ``g`` is not triangle-free, then returns ``False``.
4912
4913
EXAMPLES:
4914
4915
sage: is_maximal_triangle_free(graphs.CompleteGraph(2))
4916
True
4917
4918
sage: is_maximal_triangle_free(graphs.CycleGraph(5))
4919
True
4920
4921
sage: is_maximal_triangle_free(Graph('Esa?'))
4922
True
4923
4924
sage: is_maximal_triangle_free(Graph('KsaCCA?_C?O?'))
4925
True
4926
4927
sage: is_maximal_triangle_free(graphs.PathGraph(5))
4928
False
4929
4930
sage: is_maximal_triangle_free(Graph('LQY]?cYE_sBOE_'))
4931
False
4932
4933
sage: is_maximal_triangle_free(graphs.HouseGraph())
4934
False
4935
4936
Edge cases ::
4937
4938
sage: is_maximal_triangle_free(Graph(0))
4939
False
4940
4941
sage: is_maximal_triangle_free(Graph(1))
4942
False
4943
4944
sage: is_maximal_triangle_free(Graph(3))
4945
False
4946
"""
4947
if not g.is_triangle_free():
4948
return False
4949
g_comp = g.complement()
4950
g_copy = g.copy()
4951
for e in g_comp.edges():
4952
g_copy.add_edge(e)
4953
if g.is_triangle_free():
4954
return False
4955
g_copy.delete_edge(e)
4956
return True
4957
4958
def is_locally_two_connected(g):
4959
"""
4960
4961
ALGORITHM:
4962
4963
We modify the algorithm from our ``localise`` factory method to stop at
4964
subgraphs of 2 vertices, since ``is_two_connected`` is undefined on smaller
4965
subgraphs.
4966
"""
4967
return all((f(g.subgraph(g.neighbors(v))) if len(g.neighbors(v)) >= 2
4968
else True) for v in g.vertices())
4969
4970
######################################################################################################################
4971
#Below are some factory methods which create properties based on invariants or other properties
4972
4973
def has_equal_invariants(invar1, invar2, name=None):
4974
"""
4975
This function takes two invariants as an argument and returns the property that these invariants are equal.
4976
Optionally a name for the new function can be provided as a third argument.
4977
"""
4978
def equality_checker(g):
4979
return invar1(g) == invar2(g)
4980
4981
if name is not None:
4982
equality_checker.__name__ = name
4983
elif hasattr(invar1, '__name__') and hasattr(invar2, '__name__'):
4984
equality_checker.__name__ = 'has_{}_equals_{}'.format(invar1.__name__, invar2.__name__)
4985
else:
4986
raise ValueError('Please provide a name for the new function')
4987
4988
return equality_checker
4989
4990
"""
4991
sage: has_alpha_equals_clique_covering(graphs.CycleGraph(5))
4992
False
4993
"""
4994
has_alpha_equals_clique_covering = has_equal_invariants(independence_number, clique_covering_number, name="has_alpha_equals_clique_covering")
4995
4996
4997
def has_invariant_equal_to(invar, value, name=None, documentation=None):
4998
"""
4999
This function takes an invariant and a value as arguments and returns the property
5000
that the invariant value for a graph is equal to the provided value.
5001
5002
Optionally a name and documentation for the new function can be provided.
5003
5004
sage: order_is_five = has_invariant_equal_to(Graph.order, 5)
5005
sage: order_is_five(graphs.CycleGraph(5))
5006
True
5007
sage: order_is_five(graphs.CycleGraph(6))
5008
False
5009
"""
5010
def equality_checker(g):
5011
return invar(g) == value
5012
5013
if name is not None:
5014
equality_checker.__name__ = name
5015
elif hasattr(invar, '__name__'):
5016
equality_checker.__name__ = 'has_{}_equal_to_{}'.format(invar.__name__, value)
5017
else:
5018
raise ValueError('Please provide a name for the new function')
5019
5020
equality_checker.__doc__ = documentation
5021
5022
return equality_checker
5023
5024
def has_leq_invariants(invar1, invar2, name=None):
5025
"""
5026
This function takes two invariants as an argument and returns the property that the first invariant is
5027
less than or equal to the second invariant.
5028
Optionally a name for the new function can be provided as a third argument.
5029
"""
5030
def comparator(g):
5031
return invar1(g) <= invar2(g)
5032
5033
if name is not None:
5034
comparator.__name__ = name
5035
elif hasattr(invar1, '__name__') and hasattr(invar2, '__name__'):
5036
comparator.__name__ = 'has_{}_leq_{}'.format(invar1.__name__, invar2.__name__)
5037
else:
5038
raise ValueError('Please provide a name for the new function')
5039
5040
return comparator
5041
5042
#add all properties derived from pairs of invariants
5043
invariant_relation_properties = [has_leq_invariants(f,g) for f in all_invariants for g in all_invariants if f != g]
5044
5045
5046
def localise(f, name=None, documentation=None):
5047
"""
5048
This function takes a property (i.e., a function taking only a graph as an argument) and
5049
returns the local variant of that property. The local variant is True if the property is
5050
True for the neighbourhood of each vertex and False otherwise.
5051
"""
5052
#create a local version of f
5053
def localised_function(g):
5054
return all((f(g.subgraph(g.neighbors(v))) if g.neighbors(v) else True) for v in g.vertices())
5055
5056
#we set a nice name for the new function
5057
if name is not None:
5058
localised_function.__name__ = name
5059
elif hasattr(f, '__name__'):
5060
if f.__name__.startswith('is_'):
5061
localised_function.__name__ = 'is_locally' + f.__name__[2:]
5062
elif f.__name__.startswith('has_'):
5063
localised_function.__name__ = 'has_locally' + f.__name__[2:]
5064
else:
5065
localised_function.__name__ = 'localised_' + f.__name__
5066
else:
5067
raise ValueError('Please provide a name for the new function')
5068
5069
localised_function.__doc__ = documentation
5070
5071
return localised_function
5072
5073
is_locally_dirac = localise(is_dirac)
5074
is_locally_bipartite = localise(Graph.is_bipartite)
5075
is_locally_planar = localise(Graph.is_planar, documentation="True if the open neighborhood of each vertex v is planar")
5076
"""
5077
Tests:
5078
sage: is_locally_unicyclic(graphs.OctahedralGraph())
5079
True
5080
sage: is_locally_unicyclic(graphs.BullGraph())
5081
False
5082
"""
5083
is_locally_unicyclic = localise(is_unicyclic, documentation="""A graph is locally unicyclic if all its local subgraphs are unicyclic.
5084
5085
Tests:
5086
sage: is_locally_unicyclic(graphs.OctahedralGraph())
5087
True
5088
sage: is_locally_unicyclic(graphs.BullGraph())
5089
False
5090
""")
5091
is_locally_connected = localise(Graph.is_connected, documentation="True if the neighborhood of every vertex is connected (stronger than claw-free)")
5092
"""
5093
sage: is_local_matching(graphs.CompleteGraph(3))
5094
True
5095
sage: is_local_matching(graphs.CompleteGraph(4))
5096
False
5097
sage: is_local_matching(graphs.FriendshipGraph(5))
5098
True
5099
"""
5100
is_local_matching = localise(is_matching, name="is_local_matching", documentation="""True if the neighborhood of each vertex consists of independent edges.
5101
5102
Tests:
5103
sage: is_local_matching(graphs.CompleteGraph(3))
5104
True
5105
sage: is_local_matching(graphs.CompleteGraph(4))
5106
False
5107
sage: is_local_matching(graphs.FriendshipGraph(5))
5108
True
5109
""")
5110
5111
######################################################################################################################
5112
5113
efficiently_computable_properties = [Graph.is_regular, Graph.is_planar,
5114
Graph.is_forest, Graph.is_eulerian, Graph.is_connected, Graph.is_clique,
5115
Graph.is_circular_planar, Graph.is_chordal, Graph.is_bipartite,
5116
Graph.is_cartesian_product,Graph.is_distance_regular, Graph.is_even_hole_free,
5117
Graph.is_gallai_tree, Graph.is_line_graph, Graph.is_overfull, Graph.is_perfect,
5118
Graph.is_split, Graph.is_strongly_regular, Graph.is_triangle_free,
5119
Graph.is_weakly_chordal, is_dirac, is_ore,
5120
is_generalized_dirac, is_van_den_heuvel, is_two_connected, is_three_connected,
5121
is_lindquester, is_claw_free, Graph.has_perfect_matching, has_radius_equal_diameter,
5122
is_not_forest, is_genghua_fan, is_cubic, diameter_equals_twice_radius,
5123
is_locally_connected, matching_covered, is_locally_dirac,
5124
is_locally_bipartite, is_locally_two_connected, Graph.is_interval, has_paw,
5125
is_paw_free, has_p4, is_p4_free, has_dart, is_dart_free, has_kite, is_kite_free,
5126
has_H, is_H_free, has_residue_equals_two, order_leq_twice_max_degree,
5127
alpha_leq_order_over_two, is_factor_critical, is_independence_irreducible,
5128
has_twin, is_twin_free, diameter_equals_two, girth_greater_than_2log, Graph.is_cycle,
5129
pairs_have_unique_common_neighbor, has_star_center, is_complement_of_chordal,
5130
has_c4, is_c4_free, is_subcubic, is_quasi_regular, is_bad, has_k4, is_k4_free,
5131
is_distance_transitive, is_unicyclic, is_locally_unicyclic, has_simplical_vertex,
5132
has_exactly_two_simplical_vertices, is_two_tree, is_locally_planar,
5133
is_four_connected, is_claw_free_paw_free, has_bull, is_bull_free,
5134
is_claw_free_bull_free, has_F, is_F_free, is_oberly_sumner, is_oberly_sumner_bull,
5135
is_oberly_sumner_p4, is_matthews_sumner, chvatals_condition, is_matching, is_local_matching,
5136
has_odd_order, has_even_order, Graph.is_circulant, Graph.has_loops,
5137
Graph.is_asteroidal_triple_free, Graph.is_block_graph, Graph.is_cactus,
5138
Graph.is_cograph, Graph.is_long_antihole_free, Graph.is_long_hole_free, Graph.is_partial_cube,
5139
Graph.is_polyhedral, Graph.is_prime, Graph.is_tree, Graph.is_apex, Graph.is_arc_transitive,
5140
Graph.is_self_complementary, is_double_clique, has_fork, is_fork_free,
5141
has_empty_KE_part]
5142
5143
intractable_properties = [Graph.is_hamiltonian, Graph.is_vertex_transitive,
5144
Graph.is_edge_transitive, has_residue_equals_alpha, Graph.is_odd_hole_free,
5145
Graph.is_semi_symmetric, is_planar_transitive, is_class1,
5146
is_class2, is_anti_tutte, is_anti_tutte2, has_lovasz_theta_equals_cc,
5147
has_lovasz_theta_equals_alpha, is_chvatal_erdos, is_heliotropic_plant,
5148
is_geotropic_plant, is_traceable, is_chordal_or_not_perfect,
5149
has_alpha_residue_equal_two, is_complement_hamiltonian, is_1_tough, is_2_tough,
5150
has_two_ham_cycles, is_two_path, is_prism_hamiltonian, is_bauer, is_jung,
5151
is_weakly_pancyclic, is_pancyclic, has_two_walk, has_alpha_equals_clique_covering,
5152
Graph.is_transitively_reduced, Graph.is_half_transitive, Graph.is_line_graph,
5153
is_haggkvist_nicoghossian, is_chromatic_index_critical]
5154
5155
removed_properties = [is_pebbling_class0]
5156
5157
"""
5158
Last version of graphs packaged checked: Sage 8.2
5159
This means checked for new functions, and for any errors/changes in old functions!
5160
sage: sage.misc.banner.version_dict()['major'] < 8 or (sage.misc.banner.version_dict()['major'] == 8 and sage.misc.banner.version_dict()['minor'] <= 2)
5161
True
5162
5163
Skip Graph.is_circumscribable() and Graph.is_inscribable() because they
5164
throw errors for the vast majority of our graphs.
5165
Skip Graph.is_biconnected() in favor of our is_two_connected(), because we
5166
prefer our name, and because we disagree with their definition on K2.
5167
We define that K2 is NOT 2-connected, it is n-1 = 1 connected.
5168
Implementation of Graph.is_line_graph() is intractable, despite a theoretically efficient algorithm existing.
5169
"""
5170
sage_properties = [Graph.is_hamiltonian, Graph.is_eulerian, Graph.is_planar,
5171
Graph.is_circular_planar, Graph.is_regular, Graph.is_chordal, Graph.is_circulant,
5172
Graph.is_interval, Graph.is_gallai_tree, Graph.is_clique, Graph.is_cycle,
5173
Graph.is_transitively_reduced, Graph.is_self_complementary, Graph.is_connected,
5174
Graph.has_loops, Graph.is_asteroidal_triple_free, Graph.is_bipartite,
5175
Graph.is_block_graph, Graph.is_cactus, Graph.is_cartesian_product,
5176
Graph.is_cograph, Graph.is_distance_regular, Graph.is_edge_transitive, Graph.is_even_hole_free,
5177
Graph.is_forest, Graph.is_half_transitive, Graph.is_line_graph,
5178
Graph.is_long_antihole_free, Graph.is_long_hole_free, Graph.is_odd_hole_free,
5179
Graph.is_overfull, Graph.is_partial_cube, Graph.is_polyhedral, Graph.is_prime,
5180
Graph.is_semi_symmetric, Graph.is_split, Graph.is_strongly_regular, Graph.is_tree,
5181
Graph.is_triangle_free, Graph.is_weakly_chordal, Graph.has_perfect_matching, Graph.is_apex,
5182
Graph.is_arc_transitive]
5183
5184
#speed notes
5185
#FAST ENOUGH (tested for graphs on 140921): is_hamiltonian, is_vertex_transitive,
5186
# is_edge_transitive, has_residue_equals_alpha, is_odd_hole_free, is_semi_symmetric,
5187
# is_line_graph, is_line_graph, is_anti_tutte, is_planar_transitive
5188
#SLOW but FIXED for SpecialGraphs: is_class1, is_class2
5189
5190
properties = efficiently_computable_properties + intractable_properties
5191
properties_plus = efficiently_computable_properties + intractable_properties + invariant_relation_properties
5192
5193
5194
invariants_from_properties = [make_invariant_from_property(property) for property in properties]
5195
invariants_plus = all_invariants + invariants_from_properties
5196
5197
# weakly_chordal = weakly chordal, i.e., the graph and its complement have no induced cycle of length at least 5
5198
5199
print("loaded properties")
5200
5201
#############################################################################
5202
# End of properties section #
5203
#############################################################################
5204
# THEORY
5205
all_invariant_theorems = []
5206
all_property_theorems = []
5207
5208
alpha_upper_bounds = []
5209
alpha_lower_bounds = []
5210
5211
hamiltonian_sufficient = []
5212
5213
#####
5214
# ALPHA UPPER BOUNDS
5215
#####
5216
5217
# R. Pepper. Binding independence. Ph. D. Dissertation. University of Houston. Houston, TX, 2004.
5218
alpha_annihilation_bound = annihilation_number
5219
add_to_lists(alpha_annihilation_bound, alpha_upper_bounds, all_invariant_theorems)
5220
5221
# Nemhauser, George L., and Leslie Earl Trotter. "Vertex packings: structural properties and algorithms." Mathematical Programming 8.1 (1975): 232-248.
5222
# Nemhauser, George L., and Leslie E. Trotter. "Properties of vertex packing and independence system polyhedra." Mathematical Programming 6.1 (1974): 48-61.
5223
alpha_fractional_bound = fractional_alpha
5224
add_to_lists(alpha_fractional_bound, alpha_upper_bounds, all_invariant_theorems)
5225
5226
# D. M. Cvetkovic, M. Doob, and H. Sachs. Spectra of graphs. Academic Press, New York, 1980.
5227
alpha_cvetkovic_bound = cvetkovic
5228
add_to_lists(alpha_cvetkovic_bound, alpha_upper_bounds, all_invariant_theorems)
5229
5230
# Trivial
5231
alpha_trivial_bound = Graph.order
5232
add_to_lists(alpha_trivial_bound, alpha_upper_bounds, all_invariant_theorems)
5233
5234
# Lovasz Theta
5235
alpha_lovasz_theta_bound = Graph.lovasz_theta
5236
add_to_lists(alpha_lovasz_theta_bound, alpha_upper_bounds, all_invariant_theorems)
5237
5238
# R. Pepper. Binding independence. Ph. D. Dissertation. University of Houston. Houston, TX, 2004.
5239
def alpha_kwok_bound(g):
5240
return order(g) - (g.size()/max_degree(g))
5241
add_to_lists(alpha_kwok_bound, alpha_upper_bounds, all_invariant_theorems)
5242
5243
# P. Hansen and M. Zheng. Sharp Bounds on the order, size, and stability number of graphs. NETWORKS 23 (1993), no. 2, 99-102.
5244
def alpha_hansen_bound(g):
5245
return floor(1/2 + sqrt(1/4 + order(g)**2 - order(g) - 2*size(g)))
5246
add_to_lists(alpha_hansen_bound, alpha_upper_bounds, all_invariant_theorems)
5247
5248
# Matching Number - Folklore
5249
def alpha_matching_number_bound(g):
5250
return order(g) - matching_number(g)
5251
add_to_lists(alpha_matching_number_bound, alpha_upper_bounds, all_invariant_theorems)
5252
5253
# Min-Degree Theorm
5254
def alpha_min_degree_bound(g):
5255
return order(g) - min_degree(g)
5256
add_to_lists(alpha_min_degree_bound, alpha_upper_bounds, all_invariant_theorems)
5257
5258
# Cut Vertices Theorem
5259
# Three Bounds on the Independence Number of a Graph - C. E. Larson, R. Pepper
5260
def alpha_cut_vertices_bound(g):
5261
return (g.order() - (card_cut_vertices(g)/2) - (1/2))
5262
add_to_lists(alpha_cut_vertices_bound, alpha_upper_bounds, all_invariant_theorems)
5263
5264
# Median Degree Theorem
5265
# Three Bounds on the Independence Number of a Graph - C. E. Larson, R. Pepper
5266
def alpha_median_degree_bound(g):
5267
return (g.order() - (median_degree(g)/2) - 1/2)
5268
add_to_lists(alpha_median_degree_bound, alpha_upper_bounds, all_invariant_theorems)
5269
5270
# Godsil-Newman Upper Bound theorem
5271
# Godsil, Chris D., and Mike W. Newman. "Eigenvalue bounds for independent sets." Journal of Combinatorial Theory, Series B 98.4 (2008): 721-734.
5272
def alpha_godsil_newman_bound(g):
5273
L = max(g.laplacian_matrix().change_ring(RDF).eigenvalues())
5274
return g.order()*(L-min_degree(g))/L
5275
add_to_lists(alpha_godsil_newman_bound, alpha_upper_bounds, all_invariant_theorems)
5276
5277
# AGX Upper Bound Theorem
5278
#Aouchiche, Mustapha, Gunnar Brinkmann, and Pierre Hansen. "Variable neighborhood search for extremal graphs. 21. Conjectures and results about the independence number." Discrete Applied Mathematics 156.13 (2008): 2530-2542.
5279
def alpha_AGX_upper_bound(g):
5280
return (g.order() + max_degree(g) - ceil(2 * sqrt(g.order() - 1)))
5281
add_to_lists(alpha_AGX_upper_bound, alpha_upper_bounds, all_invariant_theorems)
5282
5283
def alpha_li_zhang_1_bound(g):
5284
"""
5285
From:
5286
A note on eigenvalue bounds for independence numbers of non-regular graphs
5287
By Yusheng Li, Zhen Zhang
5288
"""
5289
return (((max_degree(g)) - min_degree(g) - min_eigenvalue(g)) / max_degree(g)) * g.order()
5290
add_to_lists(alpha_li_zhang_1_bound, alpha_upper_bounds, all_invariant_theorems)
5291
5292
def alpha_li_zhang_2_bound(g):
5293
"""
5294
From:
5295
A note on eigenvalue bounds for independence numbers of non-regular graphs
5296
By Yusheng Li, Zhen Zhang
5297
"""
5298
return ((max_eigenvalue(g) - min_eigenvalue(g) + max_degree(g) - 2 * min_degree(g)) / (max_eigenvalue(g) - min_eigenvalue(g) + max_degree(g) - min_degree(g))) * g.order()
5299
add_to_lists(alpha_li_zhang_2_bound, alpha_upper_bounds, all_invariant_theorems)
5300
5301
def alpha_haemers_bound(g):
5302
"""
5303
From: W. Haemers, Interlacing eigenvalues and graphs, Linear Algebra Appl. 226/228 (1995) 593–616.
5304
"""
5305
return ((-max_eigenvalue(g) * min_eigenvalue(g)) / (min_degree(g)**2 - (max_eigenvalue(g) * min_eigenvalue(g)))) * g.order()
5306
add_to_lists(alpha_haemers_bound, alpha_upper_bounds, all_invariant_theorems)
5307
5308
#####
5309
# LOWER BOUNDS
5310
#####
5311
5312
# Radius Pendants Theorem
5313
# Three Bounds on the Independence Number of a Graph - C. E. Larson, R. Pepper
5314
def alpha_radius_pendants_bound(g):
5315
return (g.radius() + (card_pendants(g)/2) - 1)
5316
add_to_lists(alpha_radius_pendants_bound, alpha_lower_bounds, all_invariant_theorems)
5317
5318
# AGX Lower Bound Theorem
5319
# Aouchiche, Mustapha, Gunnar Brinkmann, and Pierre Hansen. "Variable neighborhood search for extremal graphs. 21. Conjectures and results about the independence number." Discrete Applied Mathematics 156.13 (2008): 2530-2542.
5320
def alpha_AGX_lower_bound(g):
5321
return ceil(2 * sqrt(g.order()))
5322
add_to_lists(alpha_AGX_lower_bound, alpha_lower_bounds, all_invariant_theorems)
5323
5324
def alpha_max_degree_minus_triangles_bound(g):
5325
return max_degree(g) - g.triangles_count()
5326
add_to_lists(alpha_max_degree_minus_triangles_bound, alpha_lower_bounds, all_invariant_theorems)
5327
5328
def alpha_order_brooks_bound(g):
5329
return ceil(g.order()/brooks(g))
5330
add_to_lists(alpha_order_brooks_bound, alpha_lower_bounds, all_invariant_theorems)
5331
5332
def alpha_szekeres_wilf_bound(g):
5333
return ceil(g.order()/szekeres_wilf(g))
5334
add_to_lists(alpha_szekeres_wilf_bound, alpha_lower_bounds, all_invariant_theorems)
5335
5336
def alpha_welsh_powell_bound(g):
5337
return ceil(g.order()/welsh_powell(g))
5338
add_to_lists(alpha_welsh_powell_bound, alpha_lower_bounds, all_invariant_theorems)
5339
5340
def alpha_staton_girth_bound(g):
5341
"""
5342
Hopkins, Glenn, and William Staton. "Girth and independence ratio." Can. Math. Bull. 25.2 (1982): 179-186.
5343
"""
5344
if g.girth() < 6:
5345
return 1
5346
else:
5347
d = max_degree(g)
5348
return order(g) * (2* d - 1) / (d*d + 2 * d - 1)
5349
add_to_lists(alpha_staton_girth_bound, alpha_lower_bounds, all_invariant_theorems)
5350
5351
def alpha_staton_triangle_free_bound(g):
5352
"""
5353
Staton, William. "Some Ramsey-type numbers and the independence ratio." Transactions of the American Mathematical Society 256 (1979): 353-370.
5354
"""
5355
if g.is_triangle_free() and (max_degree(g) > 2):
5356
return (5 * g.order() ) / ((5 * max_degree(g)) - 1)
5357
return 1
5358
add_to_lists(alpha_staton_triangle_free_bound, alpha_lower_bounds, all_invariant_theorems)
5359
5360
alpha_average_distance_bound = Graph.average_distance
5361
add_to_lists(alpha_average_distance_bound, alpha_lower_bounds, all_invariant_theorems)
5362
5363
alpha_radius_bound = Graph.radius
5364
add_to_lists(alpha_radius_bound, alpha_lower_bounds, all_invariant_theorems)
5365
5366
alpha_residue_bound = residue
5367
add_to_lists(alpha_residue_bound, alpha_lower_bounds, all_invariant_theorems)
5368
5369
alpha_max_even_minus_even_horizontal_bound = max_even_minus_even_horizontal
5370
add_to_lists(alpha_max_even_minus_even_horizontal_bound, alpha_lower_bounds, all_invariant_theorems)
5371
5372
alpha_critical_independence_number_bound = critical_independence_number
5373
add_to_lists(alpha_critical_independence_number_bound, alpha_lower_bounds, all_invariant_theorems)
5374
5375
def alpha_max_degree_minus_number_of_triangles_bound(g):
5376
return max_degree(g) - g.triangles_count()
5377
add_to_lists(alpha_max_degree_minus_number_of_triangles_bound, alpha_lower_bounds, all_invariant_theorems)
5378
5379
def alpha_HHRS_bound(g):
5380
"""
5381
Returns 1 if max_degree > 3 or if g has k4 as a subgraph
5382
ONLY WORKS FOR CONNECTED GRAPHS becasue that is what we are focussing on, disconnected graphs just need to count the bad compenents
5383
5384
Harant, Jochen, et al. "The independence number in graphs of maximum degree three." Discrete Mathematics 308.23 (2008): 5829-5833.
5385
"""
5386
assert(g.is_connected() == true)
5387
if not is_subcubic(g):
5388
return 1
5389
if has_k4(g):
5390
return 1
5391
return (4*g.order() - g.size() - (1 if is_bad(g) else 0) - subcubic_tr(g)) / 7
5392
add_to_lists(alpha_HHRS_bound, alpha_lower_bounds, all_invariant_theorems)
5393
5394
def alpha_seklow_bound(g):
5395
"""
5396
Returns the Seklow bound from:
5397
Selkow, Stanley M. "A probabilistic lower bound on the independence number of graphs." Discrete Mathematics 132.1-3 (1994): 363-365.
5398
"""
5399
v_sum = 0
5400
for v in g.vertices():
5401
d = g.degree(v)
5402
v_sum += ((1/(d + 1)) * (1 + max(0, (d/(d + 1) - sum([(1/(g.degree(w) + 1)) for w in g.neighbors(v)])))))
5403
return v_sum
5404
add_to_lists(alpha_seklow_bound, alpha_lower_bounds, all_invariant_theorems)
5405
5406
def alpha_harant_bound(g):
5407
"""
5408
From:
5409
A lower bound on the independence number of a graph
5410
Jochen Harant
5411
"""
5412
return (caro_wei(g)**2) / (caro_wei(g) - sum([(g.degree(e[0]) - g.degree(e[1]))**2 * (1/g.degree(e[0]))**2 * (1/g.degree(e[1]))**2 for e in g.edges()]))
5413
add_to_lists(alpha_harant_bound, alpha_lower_bounds, all_invariant_theorems)
5414
5415
def alpha_harant_schiermeyer_bound(g):
5416
"""
5417
From:
5418
On the independence number of a graph in terms of order and size
5419
By J. Harant and I. Schiermeyerb
5420
"""
5421
order = g.order()
5422
t = 2*g.size() + order + 1
5423
return (t - sqrt(t**2 - 4 * order**2)) / 2
5424
add_to_lists(alpha_harant_schiermeyer_bound, alpha_lower_bounds, all_invariant_theorems)
5425
5426
def alpha_shearer_bound(g):
5427
"""
5428
From:
5429
Shearer, James B. "The independence number of dense graphs with large odd girth." Electron. J. Combin 2.2 (1995).
5430
"""
5431
girth = g.girth()
5432
5433
if girth == +Infinity:
5434
return 1.0
5435
if is_even(girth):
5436
return 1.0
5437
5438
k = ((girth - 1) / 2.0)
5439
v_sum = sum([g.degree(v)**(1/(k - 1.0)) for v in g.vertices()])
5440
return 2**(-((k - 1.0)/k)) * v_sum**((k - 1.0)/k)
5441
add_to_lists(alpha_shearer_bound, alpha_lower_bounds, all_invariant_theorems)
5442
5443
5444
####
5445
# HAMILTONICITY SUFFICIENT CONDITIONS
5446
####
5447
# Chvátal, V., & Erdös, P. (1972). A note on hamiltonian circuits. Discrete Mathematics, 2(2), 111-113.
5448
add_to_lists(is_chvatal_erdos, hamiltonian_sufficient, all_property_theorems)
5449
5450
# R.J Faudree, Ronald J Gould, Michael S Jacobson, R.H Schelp, Neighborhood unions and hamiltonian properties in graphs, Journal of Combinatorial Theory, Series B, Volume 47, Issue 1, 1989, Pages 1-9
5451
add_to_lists(is_generalized_dirac, hamiltonian_sufficient, all_property_theorems)
5452
5453
# Häggkvist, Roland & Nicoghossian, G. G. (1981). A remark on hamiltonian cycles. Journal of Combinatorial Theory, 30(1), 118-120
5454
add_to_lists(is_haggkvist_nicoghossian, hamiltonian_sufficient, all_property_theorems)
5455
5456
# Fan, G. H. (1984). New sufficient conditions for cycles in graphs. Journal of Combinatorial Theory, 37(3), 221-227.
5457
add_to_lists(is_genghua_fan, hamiltonian_sufficient, all_property_theorems)
5458
5459
# Lindquester, T. E. (1989). The effects of distance and neighborhood union conditions on hamiltonian properties in graphs. Journal of Graph Theory, 13(3), 335-352.
5460
# Ore, Ø. (1960), "Note on Hamilton circuits", American Mathematical Monthly, 67 (1): 55, doi:10.2307/2308928, JSTOR 2308928.
5461
def is_lindquester_or_is_ore(g):
5462
return is_lindquester(g) or is_ore(g)
5463
add_to_lists(is_lindquester_or_is_ore, hamiltonian_sufficient, all_property_theorems)
5464
5465
# Trivial / "belongs to folklore"
5466
def is_cycle_or_is_clique(g):
5467
return is_cycle(g) or g.is_clique()
5468
add_to_lists(is_cycle_or_is_clique, hamiltonian_sufficient, all_property_theorems)
5469
5470
# Geng-Hua Fan. "New Sufficient Conditions for Cycles in Graphs". Journal of Combinatorial Theory 37.3(1984):221-227.
5471
def sigma_dist2_geq_half_n(g):
5472
return sigma_dist2(g) >= g.order()/2
5473
add_to_lists(sigma_dist2_geq_half_n, hamiltonian_sufficient, all_property_theorems)
5474
5475
# Bauer, Douglas, et al. "Long cycles in graphs with large degree sums." Discrete Mathematics 79.1 (1990): 59-70.
5476
add_to_lists(is_bauer, hamiltonian_sufficient, all_property_theorems)
5477
5478
# Jung, H. A. "On maximal circuits in finite graphs." Annals of Discrete Mathematics. Vol. 3. Elsevier, 1978. 129-144.
5479
add_to_lists(is_jung, hamiltonian_sufficient, all_property_theorems)
5480
5481
# S. Goodman and S. Hedetniemi, Sufficient Conditions for a Graph to Be Hamiltonian. Journal of Combinatorial Theory 16: 175--180, 1974.
5482
def is_two_connected_claw_free_paw_free(g):
5483
return is_two_connected(g) and is_claw_free_paw_free(g)
5484
add_to_lists(is_claw_free_paw_free, hamiltonian_sufficient, all_property_theorems)
5485
5486
# Ronald Gould, Updating the Hamiltonian problem — a survey. Journal of Graph Theory 15.2: 121-157, 1991.
5487
add_to_lists(is_oberly_sumner, hamiltonian_sufficient, all_property_theorems)
5488
add_to_lists(is_oberly_sumner_bull, hamiltonian_sufficient, all_property_theorems)
5489
add_to_lists(is_oberly_sumner_p4, hamiltonian_sufficient, all_property_theorems)
5490
add_to_lists(is_matthews_sumner, hamiltonian_sufficient, all_property_theorems)
5491
add_to_lists(is_broersma_veldman_gould, hamiltonian_sufficient, all_property_theorems)
5492
5493
# Chvátal, Václav. "On Hamilton's ideals." Journal of Combinatorial Theory, Series B 12.2 (1972): 163-168.
5494
add_to_lists(chvatals_condition, hamiltonian_sufficient, all_property_theorems)
5495
5496
5497
####
5498
# HAMILTONICITY NECESSARY CONDITIONS
5499
####
5500
5501
print("loaded theorems")
5502
5503
#############################################################################
5504
# End of theorems section #
5505
#############################################################################
5506
# Graph Lists
5507
5508
graph_objects = []
5509
alpha_critical_easy = []
5510
alpha_critical_hard = []
5511
chromatic_index_critical = []
5512
chromatic_index_critical_7 = []
5513
class0graphs = []
5514
class0small = []
5515
counter_examples = []
5516
problem_graphs = []
5517
sloane_graphs = []
5518
non_connected_graphs = []
5519
dimacs_graphs = []
5520
all_graphs = []
5521
5522
# HexahedralGraph is CE to (((is_planar)&(is_regular))&(is_bipartite))->(has_residue_equals_alpha)
5523
# WagnerGraph is a graph for which the Cvetkovic bound is the best upper bound present in the Willis Thesis
5524
# OctohedralGraph is a graph for which the minimum degree is the best upper bound present in the Willis thesis
5525
# BidiakisCube is a graph where none of the upper or lower bounds in the Willis thesis give the exact value for alpha
5526
# TetrahedralGraph and MoserSpindle in the alpha critical list as "C~" and "FzEKW" respectively
5527
# MeredithGraph and SchlaefliGraph are in the Problem Graphs list
5528
5529
sage_graphs = [graphs.BullGraph(), graphs.ButterflyGraph(), graphs.ClawGraph(),
5530
graphs.DiamondGraph(), graphs.HouseGraph(), graphs.HouseXGraph(), graphs.Balaban10Cage(),
5531
graphs.Balaban11Cage(), graphs.BidiakisCube(),
5532
graphs.BiggsSmithGraph(), graphs.BlanusaFirstSnarkGraph(), graphs.BlanusaSecondSnarkGraph(),
5533
graphs.BrinkmannGraph(), graphs.BrouwerHaemersGraph(), graphs.BuckyBall(),
5534
graphs.ChvatalGraph(), graphs.ClebschGraph(),
5535
graphs.CoxeterGraph(), graphs.DesarguesGraph(), graphs.DejterGraph(), graphs.DoubleStarSnark(),
5536
graphs.DurerGraph(), graphs.DyckGraph(), graphs.EllinghamHorton54Graph(),
5537
graphs.EllinghamHorton78Graph(), graphs.ErreraGraph(), graphs.F26AGraph(), graphs.FlowerSnark(),
5538
graphs.FolkmanGraph(), graphs.FosterGraph(), graphs.FranklinGraph(), graphs.FruchtGraph(),
5539
graphs.GoldnerHararyGraph(), graphs.GossetGraph(), graphs.GrayGraph(), graphs.GrotzschGraph(),
5540
graphs.HallJankoGraph(), graphs.HarborthGraph(), graphs.HarriesGraph(), graphs.HarriesWongGraph(),
5541
graphs.HeawoodGraph(), graphs.HerschelGraph(), graphs.HigmanSimsGraph(), graphs.HoffmanGraph(),
5542
graphs.HoffmanSingletonGraph(), graphs.HoltGraph(), graphs.HortonGraph(), graphs.JankoKharaghaniTonchevGraph(), graphs.KittellGraph(),
5543
graphs.KrackhardtKiteGraph(), graphs.Klein3RegularGraph(), graphs.Klein7RegularGraph(),
5544
graphs.LjubljanaGraph(),
5545
graphs.MarkstroemGraph(), graphs.McGeeGraph(),
5546
graphs.MoebiusKantorGraph(), graphs.NauruGraph(), graphs.PappusGraph(),
5547
graphs.PoussinGraph(), graphs.PerkelGraph(), graphs.PetersenGraph(), graphs.RobertsonGraph(),
5548
graphs.ShrikhandeGraph(), graphs.SimsGewirtzGraph(),
5549
graphs.SousselierGraph(), graphs.SylvesterGraph(), graphs.SzekeresSnarkGraph(),
5550
graphs.ThomsenGraph(), graphs.TietzeGraph(),
5551
graphs.TruncatedTetrahedralGraph(), graphs.Tutte12Cage(), graphs.TutteCoxeterGraph(),
5552
graphs.TutteGraph(), graphs.WagnerGraph(), graphs.WatkinsSnarkGraph(), graphs.WellsGraph(),
5553
graphs.WienerArayaGraph(),
5554
graphs.HexahedralGraph(), graphs.DodecahedralGraph(), graphs.OctahedralGraph(), graphs.IcosahedralGraph()]
5555
5556
try:
5557
add_to_lists(graphs.CameronGraph(), sage_graphs)
5558
except Exception as e:
5559
print("The Cameron graph was not loaded. Caused by:")
5560
print(e)
5561
5562
try:
5563
add_to_lists(graphs.M22Graph(), sage_graphs)
5564
except Exception as e:
5565
print("The M22 graph was not loaded. Caused by:")
5566
print(e)
5567
5568
try:
5569
add_to_lists(graphs.TruncatedIcosidodecahedralGraph(), sage_graphs)
5570
except Exception as e:
5571
print("The truncated icosidodecahedral graph was not loaded. Caused by:")
5572
print(e)
5573
5574
try:
5575
add_to_lists(graphs.McLaughlinGraph(), sage_graphs)
5576
except Exception as e:
5577
print("The McLaughlin graph was not loaded. Caused by:")
5578
print(e)
5579
5580
try:
5581
add_to_lists(graphs.LocalMcLaughlinGraph(), sage_graphs)
5582
except Exception as e:
5583
print("The local McLaughlin graph was not loaded. Caused by:")
5584
print(e)
5585
5586
#hard built-in Sage graphs
5587
add_to_lists(graphs.IoninKharaghani765Graph(), problem_graphs, all_graphs)
5588
5589
5590
#These built in graphs are nameless so here they are given names
5591
5592
try:
5593
cell120 = graphs.Cell120()
5594
cell120.name(new = "Cell120")
5595
add_to_lists(cell120, sage_graphs)
5596
except Exception as e:
5597
print("The graph of the 120-cell was not loaded. Caused by:")
5598
print(e)
5599
5600
try:
5601
cell600 = graphs.Cell600()
5602
cell600.name(new = "Cell600")
5603
add_to_lists(cell600, sage_graphs)
5604
except Exception as e:
5605
print("The graph of the 600-cell was not loaded. Caused by:")
5606
print(e)
5607
5608
mathon_strongly_regular0 = graphs.MathonStronglyRegularGraph(0)
5609
mathon_strongly_regular0.name(new = "Mathon Strongly Regular Graph 0")
5610
5611
mathon_strongly_regular1 = graphs.MathonStronglyRegularGraph(1)
5612
mathon_strongly_regular1.name(new = "Mathon Strongly Regular Graph 1")
5613
5614
mathon_strongly_regular2 = graphs.MathonStronglyRegularGraph(2)
5615
mathon_strongly_regular2.name(new = "Mathon Strongly Regular Graph 2")
5616
5617
janko_kharaghani936 = graphs.JankoKharaghaniGraph(936)
5618
janko_kharaghani936.name(new = "Janko-Kharaghani 936")
5619
5620
# janko_kharaghani1800 = graphs.JankoKharaghaniGraph(1800) # Causes memory error when doctesting. Waiting for fix in Sage.
5621
# janko_kharaghani1800.name(new = "Janko-Kharagani 1800")
5622
5623
for graph in sage_graphs + [mathon_strongly_regular0, mathon_strongly_regular1, mathon_strongly_regular2, janko_kharaghani936]:
5624
add_to_lists(graph, graph_objects, all_graphs)
5625
5626
# Meredith graph is 4-reg, class2, non-hamiltonian: http://en.wikipedia.org/wiki/Meredith_graph
5627
add_to_lists(graphs.MeredithGraph(), problem_graphs, all_graphs)
5628
add_to_lists(graphs.SchlaefliGraph(), problem_graphs, all_graphs)
5629
5630
# A graph is alpha_critical if removing any edge increases independence number
5631
# All alpha critical graphs of orders 2 to 9, 53 in total
5632
5633
# "E|OW" is CE to (has_alpha_residue_equal_two)->((is_perfect)|(is_regular))
5634
5635
alpha_critical_graph_names = ['A_','Bw', 'C~', 'Dhc', 'D~{', 'E|OW', 'E~~w', 'FhCKG', 'F~[KG',
5636
'FzEKW', 'Fn[kG', 'F~~~w', 'GbL|TS', 'G~?mvc', 'GbMmvG', 'Gb?kTG', 'GzD{Vg', 'Gb?kR_', 'GbqlZ_',
5637
'GbilZ_', 'G~~~~{', 'GbDKPG', 'HzCGKFo', 'H~|wKF{', 'HnLk]My', 'HhcWKF_', 'HhKWKF_', 'HhCW[F_',
5638
'HxCw}V`', 'HhcGKf_', 'HhKGKf_', 'Hh[gMEO', 'HhdGKE[', 'HhcWKE[', 'HhdGKFK', 'HhCGGE@', 'Hn[gGE@',
5639
'Hn^zxU@', 'HlDKhEH', 'H~~~~~~', 'HnKmH]N', 'HnvzhEH', 'HhfJGE@', 'HhdJGM@', 'Hj~KHeF', 'HhdGHeB',
5640
'HhXg[EO', 'HhGG]ES', 'H~Gg]f{', 'H~?g]vs', 'H~@w[Vs', 'Hn_k[^o']
5641
5642
for s in alpha_critical_graph_names:
5643
g = Graph(s)
5644
g.name(new="alpha_critical_"+ s)
5645
add_to_lists(g, alpha_critical_easy, graph_objects, all_graphs)
5646
5647
# All order-7 chromatic_index_critical_graphs (and all are overfull)
5648
n7_chromatic_index_critical_names = ['FhCKG', 'FzCKW', 'FzNKW', 'FlSkG', 'Fn]kG', 'FlLKG', 'FnlkG', 'F~|{G', 'FnlLG', 'F~|\\G',
5649
'FnNLG', 'F~^LW', 'Fll\\G', 'FllNG', 'F~l^G', 'F~|^w', 'F~~^W', 'Fnl^W', 'FlNNG', 'F|\\Kg',
5650
'F~^kg', 'FlKMG']
5651
5652
for s in n7_chromatic_index_critical_names:
5653
g=Graph(s)
5654
g.name(new="chromatic_index_critical_7_" + s)
5655
add_to_lists(g, chromatic_index_critical, chromatic_index_critical_7, problem_graphs, all_graphs)
5656
5657
alpha_critical_hard = [Graph('Hj\\x{F{')]
5658
for g in alpha_critical_hard:
5659
add_to_lists(g, problem_graphs, all_graphs)
5660
5661
# Graph objects
5662
5663
p3 = graphs.PathGraph(3)
5664
p3.name(new = "p3")
5665
add_to_lists(p3, graph_objects, all_graphs)
5666
5667
p4 = graphs.PathGraph(4)
5668
p4.name(new="p4")
5669
add_to_lists(p4, graph_objects, all_graphs)
5670
5671
p5 = graphs.PathGraph(5)
5672
p5.name(new = "p5")
5673
add_to_lists(p5, graph_objects, all_graphs)
5674
5675
p6 = graphs.PathGraph(6)
5676
p6.name(new="p6")
5677
add_to_lists(p6, graph_objects, all_graphs)
5678
5679
"""
5680
CE to independence_number(x) <= e^(cosh(max_degree(x) - 1))
5681
and to
5682
independence_number(x) <= max_degree(x)*min_degree(x) + card_periphery(x)
5683
"""
5684
p9 = graphs.PathGraph(9)
5685
p9.name(new = "p9")
5686
add_to_lists(p9, graph_objects, counter_examples, all_graphs)
5687
5688
"""
5689
P29 is a CE to independence_number(x) <=degree_sum(x)/sqrt(card_negative_eigenvalues(x))
5690
and to
5691
<= max_degree(x)^e^card_center(x)
5692
and to
5693
<= max_degree(x)^2 + card_periphery(x)
5694
"""
5695
p29 = graphs.PathGraph(29)
5696
p29.name(new = "p29")
5697
add_to_lists(p29, graph_objects, counter_examples, all_graphs)
5698
5699
# CE to independence_number(x) <= 2*cvetkovic(x)*log(10)/log(x.size())
5700
p102 = graphs.PathGraph(102)
5701
p102.name(new = "p102")
5702
add_to_lists(p102, graph_objects, counter_examples, all_graphs)
5703
5704
# CE to independence_number(x)<=welsh_powell(x)^e^different_degrees(x)
5705
p6707 = graphs.PathGraph(6707)
5706
p6707.name(new = "p6707")
5707
5708
c4 = graphs.CycleGraph(4)
5709
c4.name(new="c4")
5710
add_to_lists(c4, graph_objects, all_graphs)
5711
5712
c6 = graphs.CycleGraph(6)
5713
c6.name(new = "c6")
5714
add_to_lists(c6, graph_objects, all_graphs)
5715
5716
# CE to independence_number(x) <= (e^welsh_powell(x) - graph_rank(x))^2
5717
c22 = graphs.CycleGraph(22)
5718
c22.name(new = "c22")
5719
add_to_lists(c22, graph_objects, counter_examples, all_graphs)
5720
5721
# CE to independence_number(x) <= minimum(cvetkovic(x), 2*e^sum_temperatures(x))
5722
c34 = graphs.CycleGraph(34)
5723
c34.name(new = "c34")
5724
add_to_lists(c34, graph_objects, counter_examples, all_graphs)
5725
5726
# CE to independence_number(x) <= residue(x)^(degree_sum(x)^density(x))
5727
c102 = graphs.CycleGraph(102)
5728
c102.name(new = "c102")
5729
add_to_lists(c102, graph_objects, counter_examples, all_graphs)
5730
5731
"""
5732
Sage defines circulant graphs without the cycle, whereas the paper defines it with the cycle
5733
From:
5734
Brimkov, Valentin. "Algorithmic and explicit determination of the Lovasz number for certain circulant graphs." Discrete Applied Mathematics 155.14 (2007): 1812-1825.
5735
"""
5736
c13_2 = graphs.CirculantGraph(13, 2)
5737
c13_2.add_cycle(range(13))
5738
c13_2.name(new = "c13_2")
5739
add_to_lists(c13_2, graph_objects, all_graphs)
5740
5741
k3 = alpha_critical_easy[1]
5742
k4 = alpha_critical_easy[2]
5743
5744
k10 = graphs.CompleteGraph(10)
5745
k10.name(new="k10")
5746
add_to_lists(k10, graph_objects, all_graphs)
5747
5748
# CE to independence_number(x) >= floor(tan(floor(gutman_energy(x))))
5749
k37 = graphs.CompleteGraph(37)
5750
k37.name(new = "k37")
5751
add_to_lists(k37, graph_objects, counter_examples, all_graphs)
5752
5753
# Lipták, László, and László Lovász. "Critical facets of the stable set polytope." Combinatorica 21.1 (2001): 61-88.
5754
k1_4 = graphs.CompleteBipartiteGraph(1, 4)
5755
k1_4.name(new = "k1_4")
5756
add_to_lists(k1_4, graph_objects, all_graphs)
5757
5758
# A 4 ray star with each ray subdivided into two edges
5759
# Lipták, László, and László Lovász. "Critical facets of the stable set polytope." Combinatorica 21.1 (2001): 61-88.
5760
k1_4_sub2 = Graph("H?EA@?^")
5761
k1_4_sub2.name(new = "k1_4_sub2")
5762
add_to_lists(k1_4_sub2, graph_objects, all_graphs)
5763
5764
# CE to independence_number(x) <= minimum(lovasz_theta(x), 2*e^sum_temperatures(x))
5765
# and to
5766
# independence_number(x) <= minimum(floor(lovasz_theta(x)), 2*e^sum_temperatures(x))
5767
# and to
5768
# independence_number(x) >= -brinkmann_steffen(x) + 1/2*card_center(x)
5769
k1_9 = graphs.CompleteBipartiteGraph(1,9)
5770
k1_9.name(new = "k1_9")
5771
add_to_lists(k1_9, graph_objects, counter_examples, all_graphs)
5772
5773
# The line graph of k3,3
5774
k3_3_line_graph = graphs.CompleteBipartiteGraph(3, 3).line_graph()
5775
k3_3_line_graph.name(new = "k3_3 line graph")
5776
add_to_lists(k3_3_line_graph, graph_objects, all_graphs)
5777
5778
k5_3=graphs.CompleteBipartiteGraph(5,3)
5779
k5_3.name(new = "k5_3")
5780
add_to_lists(k5_3, graph_objects, all_graphs)
5781
5782
# CE to independence_number(x) <= diameter^(max_degree-1)
5783
# diameter is 16, Delta=3, alpha = 341
5784
bt2_8 = graphs.BalancedTree(2,8)
5785
bt2_8.name(new = "bt2_8")
5786
add_to_lists(bt2_8, graph_objects, counter_examples, all_graphs)
5787
5788
#two c4's joined at a vertex
5789
c4c4=graphs.CycleGraph(4)
5790
for i in [4,5,6]:
5791
c4c4.add_vertex()
5792
c4c4.add_edge(3,4)
5793
c4c4.add_edge(5,4)
5794
c4c4.add_edge(5,6)
5795
c4c4.add_edge(6,3)
5796
c4c4.name(new="c4c4")
5797
add_to_lists(c4c4, graph_objects, all_graphs)
5798
5799
#two c5's joined at a vertex: eulerian, not perfect, not hamiltonian
5800
c5c5=graphs.CycleGraph(5)
5801
for i in [5,6,7,8]:
5802
c5c5.add_vertex()
5803
c5c5.add_edge(0,5)
5804
c5c5.add_edge(0,8)
5805
c5c5.add_edge(6,5)
5806
c5c5.add_edge(6,7)
5807
c5c5.add_edge(7,8)
5808
c5c5.name(new="c5c5")
5809
add_to_lists(c5c5, graph_objects, all_graphs)
5810
5811
K4a=graphs.CompleteGraph(4)
5812
K4b=graphs.CompleteGraph(4)
5813
K4a.delete_edge(0,1)
5814
K4b.delete_edge(0,1)
5815
regular_non_trans = K4a.disjoint_union(K4b)
5816
regular_non_trans.add_edge((0,0),(1,1))
5817
regular_non_trans.add_edge((0,1),(1,0))
5818
regular_non_trans.name(new="regular_non_trans")
5819
add_to_lists(regular_non_trans, graph_objects, all_graphs)
5820
5821
c6ee = graphs.CycleGraph(6)
5822
c6ee.add_edges([(1,5), (2,4)])
5823
c6ee.name(new="c6ee")
5824
add_to_lists(c6ee, graph_objects, all_graphs)
5825
5826
#c6ee plus another chord: hamiltonian, regular, vertex transitive
5827
c6eee = copy(c6ee)
5828
c6eee.add_edge(0,3)
5829
c6eee.name(new="c6eee")
5830
add_to_lists(c6eee, graph_objects, all_graphs)
5831
5832
#c8 plus one long vertical chord and 3 parallel horizontal chords
5833
c8chorded = graphs.CycleGraph(8)
5834
c8chorded.add_edge(0,4)
5835
c8chorded.add_edge(1,7)
5836
c8chorded.add_edge(2,6)
5837
c8chorded.add_edge(3,5)
5838
c8chorded.name(new="c8chorded")
5839
add_to_lists(c8chorded, graph_objects, all_graphs)
5840
5841
#c8 plus 2 parallel chords: hamiltonian, tri-free, not vertex-transitive
5842
c8chords = graphs.CycleGraph(8)
5843
c8chords.add_edge(1,6)
5844
c8chords.add_edge(2,5)
5845
c8chords.name(new="c8chords")
5846
add_to_lists(c8chords, graph_objects, all_graphs)
5847
5848
# C10 with an edge through the center
5849
c10e = graphs.CycleGraph(10)
5850
c10e.add_edge(0,5)
5851
c10e.name(new = "c10e")
5852
add_to_lists(c10e, graph_objects, all_graphs)
5853
5854
# C10e with another edge through center
5855
c10ee = graphs.CycleGraph(10)
5856
c10ee.add_edges([(0,5), (9, 4)])
5857
c10ee.name(new = "c10ee")
5858
add_to_lists(c10ee, graph_objects, all_graphs)
5859
5860
prismsub = graphs.CycleGraph(6)
5861
prismsub.add_edge(0,2)
5862
prismsub.add_edge(3,5)
5863
prismsub.add_edge(1,4)
5864
prismsub.subdivide_edge(1,4,1)
5865
prismsub.name(new="prismsub")
5866
add_to_lists(prismsub, graph_objects, all_graphs)
5867
5868
# ham, not vertex trans, tri-free, not cartesian product
5869
prismy = graphs.CycleGraph(8)
5870
prismy.add_edge(2,5)
5871
prismy.add_edge(0,3)
5872
prismy.add_edge(4,7)
5873
prismy.name(new="prismy")
5874
add_to_lists(prismy, graph_objects, all_graphs)
5875
5876
#c10 with chords, ham, tri-free, regular, planar, vertex transitive
5877
sixfour = graphs.CycleGraph(10)
5878
sixfour.add_edge(1,9)
5879
sixfour.add_edge(0,2)
5880
sixfour.add_edge(3,8)
5881
sixfour.add_edge(4,6)
5882
sixfour.add_edge(5,7)
5883
sixfour.name(new="sixfour")
5884
add_to_lists(sixfour, graph_objects, all_graphs)
5885
5886
#unique 24-vertex fullerene: hamiltonian, planar, not vertex transitive
5887
fullerene_24 = Graph('WsP@H?PC?O`?@@?_?GG@??CC?G??GG?E???o??B???E???F')
5888
fullerene_24.name(new="fullerene_24")
5889
add_to_lists(fullerene_24, graph_objects, all_graphs)
5890
5891
#unique 26-atom fullerene: hamiltonian, planar, not vertex trans, radius=5, diam=6
5892
fullerene_26 = Graph('YsP@H?PC?O`?@@?_?G?@??CC?G??GG?E??@_??K???W???W???H???E_')
5893
fullerene_26.name(new="fullerene_26")
5894
add_to_lists(fullerene_26, graph_objects, all_graphs)
5895
5896
#the unique 100-atom fullerene with minimum independence number of 43 (and IPR, tetrahedral symmetry)
5897
fullerene_100 = Graph("~?@csP@@?OC?O`?@?@_?O?A??W??_??_G?O??C??@_??C???G???G@??K???A????O???@????A????A?G??B?????_????C?G???O????@_?????_?????O?????C?G???@_?????E??????G??????G?G????C??????@???????G???????o??????@???????@????????_?_?????W???????@????????C????????G????????G?G??????E????????@_????????K?????????_????????@?@???????@?@???????@_?????????G?????????@?@????????C?C????????W??????????W??????????C??????????@?@?????????G???????????_??????????@?@??????????_???????????O???????????C?G??????????O???????????@????????????A????????????A?G??????????@_????????????W????????????@_????????????E?????????????E?????????????E?????????????B??????????????O?????????????A@?????????????G??????????????OG?????????????O??????????????GC?????????????A???????????????OG?????????????@?_?????????????B???????????????@_???????????????W???????????????@_???????????????F")
5898
fullerene_100.name(new="fullerene_100")
5899
add_to_lists(fullerene_100, graph_objects, all_graphs)
5900
5901
"""
5902
The Holton-McKay graph is the smallest planar cubic hamiltonian graph with an edge
5903
that is not contained in a hamiltonian cycle. It has 24 vertices and the edges (0,3)
5904
and (4,7) are not contained in a hamiltonian cycle. This graph was mentioned in
5905
D. A. Holton and B. D. McKay, Cycles in 3-connected cubic planar graphs II, Ars
5906
Combinatoria, 21A (1986) 107-114.
5907
5908
sage: holton_mckay
5909
holton_mckay: Graph on 24 vertices
5910
sage: holton_mckay.is_planar()
5911
True
5912
sage: holton_mckay.is_regular()
5913
True
5914
sage: max(holton_mckay.degree())
5915
3
5916
sage: holton_mckay.is_hamiltonian()
5917
True
5918
sage: holton_mckay.radius()
5919
4
5920
sage: holton_mckay.diameter()
5921
6
5922
"""
5923
holton_mckay = Graph('WlCGKS??G?_D????_?g?DOa?C?O??G?CC?`?G??_?_?_??L')
5924
holton_mckay.name(new="holton_mckay")
5925
add_to_lists(holton_mckay, graph_objects, all_graphs)
5926
5927
#an example of a bipartite, 1-tough, not van_den_heuvel, not hamiltonian graph
5928
kratsch_lehel_muller = graphs.PathGraph(12)
5929
kratsch_lehel_muller.add_edge(0,5)
5930
kratsch_lehel_muller.add_edge(6,11)
5931
kratsch_lehel_muller.add_edge(4,9)
5932
kratsch_lehel_muller.add_edge(1,10)
5933
kratsch_lehel_muller.add_edge(2,7)
5934
kratsch_lehel_muller.name(new="kratsch_lehel_muller")
5935
add_to_lists(kratsch_lehel_muller, graph_objects, all_graphs)
5936
5937
#ham, not planar, not anti_tutte
5938
c6xc6 = graphs.CycleGraph(6).cartesian_product(graphs.CycleGraph(6))
5939
c6xc6.name(new="c6xc6")
5940
add_to_lists(c6xc6, graph_objects, all_graphs)
5941
5942
c7xc7 = graphs.CycleGraph(7).cartesian_product(graphs.CycleGraph(7))
5943
c7xc7.name(new="c7xc7")
5944
add_to_lists(c7xc7, graph_objects, all_graphs)
5945
5946
# Product Graphs, fig. 1.13
5947
c6xk2 = graphs.CycleGraph(6).cartesian_product(graphs.CompleteGraph(2))
5948
c6xk2.name(new = "c6xk2")
5949
add_to_lists(c6xk2, graph_objects, all_graphs)
5950
5951
# Product Graphs, fig. 1.13
5952
k1_4xp3 = graphs.CompleteBipartiteGraph(1, 4).cartesian_product(graphs.PathGraph(3))
5953
k1_4xp3.name(new = "k1_4xp3")
5954
add_to_lists(k1_4xp3, graph_objects, all_graphs)
5955
5956
# Product Graphs, fig. 1.14
5957
p4xk3xk2 = graphs.PathGraph(4).cartesian_product(graphs.CompleteGraph(3)).cartesian_product(graphs.CompleteGraph(2))
5958
p4xk3xk2.name(new = "p4xk3xk2")
5959
add_to_lists(p4xk3xk2, graph_objects, all_graphs)
5960
5961
# Product Graphs, fig. 4.1
5962
p3xk2xk2 = graphs.PathGraph(3).cartesian_product(graphs.CompleteGraph(2)).cartesian_product(graphs.CompleteGraph(2))
5963
p3xk2xk2.name(new = "p3xk2xk2")
5964
add_to_lists(p3xk2xk2, graph_objects, all_graphs)
5965
5966
# Product Graphs, fig. 5.1
5967
p4Xp5 = graphs.PathGraph(4).strong_product(graphs.PathGraph(5))
5968
p4Xp5.name(new = "p4Xp5")
5969
add_to_lists(p4Xp5, graph_objects, all_graphs)
5970
5971
# Product Graphs, Fig 6.1
5972
k3lxp3 = graphs.CompleteGraph(3).lexicographic_product(graphs.PathGraph(3))
5973
k3lxp3.name(new = "k3lxp3")
5974
add_to_lists(k3lxp3, graph_objects, all_graphs)
5975
5976
# Product Graphs, Fig 6.1
5977
p3lxk3 = graphs.PathGraph(3).lexicographic_product(graphs.CompleteGraph(3))
5978
p3lxk3.name(new = "p3lxk3")
5979
add_to_lists(p3lxk3, graph_objects, all_graphs)
5980
5981
"""
5982
Referenced p.15
5983
Mathew, K. Ashik, and Patric RJ Östergård. "New lower bounds for the Shannon capacity of odd cycles." Designs, Codes and Cryptography (2015): 1-10.
5984
"""
5985
c5Xc5 = graphs.CycleGraph(5).strong_product(graphs.CycleGraph(5))
5986
c5Xc5.name(new = "c5Xc5")
5987
add_to_lists(c5Xc5, graph_objects, all_graphs)
5988
5989
#non-ham, 2-connected, eulerian (4-regular)
5990
gould = Graph('S~dg?CB?wC_L????_?W?F??c?@gOOOGGK')
5991
gould.name(new="gould")
5992
add_to_lists(gould, graph_objects, all_graphs)
5993
5994
#two k5s with single edge removed from each and lines joining these 4 points to a new center point, non-hamiltonian
5995
throwing = Graph('J~wWGGB?wF_')
5996
throwing.name(new="throwing")
5997
add_to_lists(throwing, graph_objects, all_graphs)
5998
5999
#k4 plus k2 on one side, open k5 on other, meet at single point in center, non-hamiltonian
6000
throwing2 = Graph("K~wWGKA?gB_N")
6001
throwing2.name(new="throwing2")
6002
add_to_lists(throwing2, graph_objects, all_graphs)
6003
6004
#similar to throwing2 with pair of edges swapped, non-hamiltonian
6005
throwing3 = Graph("K~wWGGB?oD_N")
6006
throwing3.name(new="throwing3")
6007
add_to_lists(throwing3, graph_objects, all_graphs)
6008
6009
#graph has diameter != radius but is hamiltonian
6010
tent = graphs.CycleGraph(4).join(Graph(1),labels="integers")
6011
tent.name(new="tent")
6012
add_to_lists(tent, graph_objects, all_graphs)
6013
6014
# C5 with chords from one vertex to other 2 (showed up in auto search for CE's): hamiltonian
6015
bridge = Graph("DU{")
6016
bridge.name(new="bridge")
6017
add_to_lists(bridge, graph_objects, all_graphs)
6018
6019
# nico found the smallest hamiltonian overfull graph
6020
non_ham_over = Graph("HCQRRQo")
6021
non_ham_over.name(new="non_ham_over")
6022
add_to_lists(non_ham_over, graph_objects, all_graphs)
6023
6024
# From Ryan Pepper
6025
ryan = Graph("WxEW?CB?I?_R????_?W?@?OC?AW???O?C??B???G?A?_??R")
6026
ryan.name(new="ryan")
6027
add_to_lists(ryan, graph_objects, all_graphs)
6028
6029
# Ryan Pepper
6030
# CE to independence_number(x) <= 2 * x.chromatic_number() + 2 * residue(x)
6031
# has alpha=25,chi=2,residue=10
6032
ryan2=graphs.CirculantGraph(50,[1,3])
6033
ryan2.name(new="circulant_50_1_3")
6034
add_to_lists(ryan2, graph_objects, counter_examples, all_graphs)
6035
6036
# From Ryan Pepper
6037
# CE to independence_number(x) >= diameter(x) - 1 for regular graphs
6038
pepper_1_gadget = Graph('Ot???CA?WB`_B@O_B_B?A')
6039
pepper_1_gadget.name(new = "pepper_1_gadget")
6040
add_to_lists(pepper_1_gadget, graph_objects, counter_examples, all_graphs)
6041
6042
# p10 joined to 2 points of k4
6043
# CE to chromatic_number <= avg_degree + 1
6044
p10k4=Graph('MhCGGC@?G?_@_B?B_')
6045
p10k4.name(new="p10k4")
6046
add_to_lists(p10k4, graph_objects, counter_examples, all_graphs)
6047
6048
# star on 13 points with added edge:
6049
# CE to independence_number(x) <= dom + girth(x)^2
6050
s13e = Graph('M{aCCA?_C?O?_?_??')
6051
s13e.name(new="s13e")
6052
add_to_lists(s13e, graph_objects, counter_examples, all_graphs)
6053
6054
# CE to independence_number(x) <= 2 * girth(x)^2 + 2
6055
# Star with 22 rays plus extra edge
6056
s22e = graphs.StarGraph(22)
6057
s22e.add_edge(1,2)
6058
s22e.name(new="s22e")
6059
add_to_lists(s22e, graph_objects, counter_examples, all_graphs)
6060
6061
# Graph from delavina's jets paper
6062
starfish = Graph('N~~eeQoiCoM?Y?U?F??')
6063
starfish.name(new="starfish")
6064
add_to_lists(starfish, graph_objects, all_graphs)
6065
6066
# difficult graph from INP: order=11, alpha=4, best lower bound < 3
6067
difficult11 = Graph('J?`FBo{fdb?')
6068
difficult11.name(new="difficult11")
6069
add_to_lists(difficult11, graph_objects, all_graphs)
6070
6071
# c4 joined to K# at point: not KE, alpha=theta=nu=3, delting any vertex gives KE graph
6072
c5k3=Graph('FheCG')
6073
c5k3.name(new="c5k3")
6074
add_to_lists(c5k3, graph_objects, all_graphs)
6075
6076
# mycielskian of a triangle:
6077
# CE to chi <= max(clique, nu)
6078
# chi=4, nu = clique = 3
6079
c3mycielski = Graph('FJnV?')
6080
c3mycielski.name(new="c3mycieski")
6081
add_to_lists(c3mycielski, problem_graphs , counter_examples, all_graphs)
6082
6083
# 4th mycielskian of a triangle,
6084
# CE to chi <= clique + girth
6085
# chi = 7, clique = girth = 3
6086
c3mycielski4 = Graph('~??~??GWkYF@BcuIsJWEo@s?N?@?NyB`qLepJTgRXkAkU?JPg?VB_?W[??Ku??BU_??ZW??@u???Bs???Bw???A??F~~_B}?^sB`o[MOuZErWatYUjObXkZL_QpWUJ?CsYEbO?fB_w[?A`oCM??DL_Hk??DU_Is??Al_Dk???l_@k???Ds?M_???V_?{????oB}?????o[M?????WuZ?????EUjO?????rXk?????BUJ??????EsY??????Ew[??????B`o???????xk???????FU_???????\\k????????|_????????}_????????^_?????????')
6087
c3mycielski4.name(new="c3mycielski4")
6088
add_to_lists(c3mycielski4, graph_objects, counter_examples, all_graphs)
6089
6090
# A PAW is a traingle with a pendant
6091
# Shows up in a sufficient condition for hamiltonicity
6092
paw=Graph('C{')
6093
paw.name(new="paw")
6094
add_to_lists(paw, graph_objects, all_graphs)
6095
6096
# 2 octahedrons, remove one edge from each, add vertex, connect it to deleted edge vertices
6097
# its regular of degree 4
6098
binary_octahedron = Graph('L]lw??B?oD_Noo')
6099
binary_octahedron.name(new = "binary_octahedron")
6100
add_to_lists(binary_octahedron, graph_objects, all_graphs)
6101
6102
# this graph shows that the cartesian product of 2 KE graphs is not necessarily KE
6103
# appears in Abay-Asmerom, Ghidewon, et al. "Notes on the independence number in the Cartesian product of graphs." Discussiones Mathematicae Graph Theory 31.1 (2011): 25-35.
6104
paw_x_paw = paw.cartesian_product(paw)
6105
paw_x_paw.name(new = "paw_x_paw")
6106
add_to_lists(paw_x_paw, graph_objects, all_graphs)
6107
6108
#a DART is a kite with a pendant
6109
dart = Graph('DnC')
6110
dart.name(new="dart")
6111
add_to_lists(dart, graph_objects, all_graphs)
6112
6113
# CE to ((is_chordal)^(is_forest))->(has_residue_equals_alpha)
6114
ce2=Graph("HdGkCA?")
6115
ce2.name(new = "ce2")
6116
add_to_lists(ce2, graph_objects, counter_examples, all_graphs)
6117
6118
# CE to ((~(is_planar))&(is_chordal))->(has_residue_equals_alpha)
6119
ce4=Graph("G~sNp?")
6120
ce4.name(new = "ce4")
6121
add_to_lists(ce4, graph_objects, counter_examples, all_graphs)
6122
6123
# CE to (((is_line_graph)&(is_cartesian_product))|(is_split))->(has_residue_equals_alpha)
6124
ce5=Graph("X~}AHKVB{GGPGRCJ`B{GOO`C`AW`AwO`}CGOO`AHACHaCGVACG^")
6125
ce5.name(new = "ce5")
6126
add_to_lists(ce5, graph_objects, counter_examples, all_graphs)
6127
6128
# CE to (is_split)->((order_leq_twice_max_degree)&(is_chordal))
6129
ce6 = Graph("H??E@cN")
6130
ce6.name(new = "ce6")
6131
add_to_lists(ce6, graph_objects, counter_examples, all_graphs)
6132
6133
# CE to (has_residue_equals_alpha)->((is_bipartite)->(order_leq_twice_max_degree))
6134
ce7 = Graph("FpGK?")
6135
ce7.name(new = "ce7")
6136
add_to_lists(ce7, graph_objects, counter_examples, all_graphs)
6137
6138
# CE to ((has_paw)&(is_circular_planar))->(has_residue_equals_alpha)
6139
ce8 = Graph('IxCGGC@_G')
6140
ce8.name(new = "ce8")
6141
add_to_lists(ce8, graph_objects, counter_examples, all_graphs)
6142
6143
# CE to ((has_H)&(is_forest))->(has_residue_equals_alpha)
6144
ce9 = Graph('IhCGGD?G?')
6145
ce9.name(new = "ce9")
6146
add_to_lists(ce9, graph_objects, counter_examples, all_graphs)
6147
6148
# CE to (((is_eulerian)&(is_planar))&(has_paw))->(has_residue_equals_alpha)
6149
ce10=Graph('KxkGGC@?G?o@')
6150
ce10.name(new = "ce10")
6151
add_to_lists(ce10, graph_objects, counter_examples, all_graphs)
6152
6153
# CE to (((is_cubic)&(is_triangle_free))&(is_H_free))->(has_residue_equals_two)
6154
ce12 = Graph("Edo_")
6155
ce12.name(new = "ce12")
6156
add_to_lists(ce12, graph_objects, counter_examples, all_graphs)
6157
6158
# CE to ((diameter_equals_twice_radius)&(is_claw_free))->(has_residue_equals_two)
6159
ce13 = Graph("ExOG")
6160
ce13.name(new = "ce13")
6161
add_to_lists(ce13, graph_objects, counter_examples, all_graphs)
6162
6163
# CE to (~(matching_covered))->(has_residue_equals_alpha)
6164
ce14 = Graph('IhCGGC_@?')
6165
ce14.name(new = "IhCGGC_@?")
6166
add_to_lists(ce14, graph_objects, counter_examples, all_graphs)
6167
6168
"""
6169
CE to independence_number(x) <= 10^order_automorphism_group(x)
6170
6171
sage: order(ce15)
6172
57
6173
sage: independence_number(ce15)
6174
25
6175
"""
6176
ce15 = Graph("x??C?O?????A?@_G?H??????A?C??EGo?@S?O@?O??@G???CO???CAC_??a?@G?????H???????????O?_?H??G??G??@??_??OA?OCHCO?YA????????A?O???G?O?@????OOC???_@??????MCOC???O_??[Q??@???????O??_G?P?GO@A?G_???A???A@??g???W???@CG_???`_@O??????@?O@?AGO?????C??A??F??????@C????A?E@L?????P@`??")
6177
ce15.name(new = "ce15")
6178
add_to_lists(ce15, graph_objects, counter_examples, all_graphs)
6179
6180
# CE to independence_number(x) <= 2*maximum(welsh_powell(x), max_even_minus_even_horizontal(x))
6181
ce16 = Graph("mG???GP?CC?Aa?GO?o??I??c??O??G?ACCGW@????OC?G@?_A_W_OC@??@?I??O?_AC?Oo?E@_?O??I??B_?@_A@@@??O?OC?GC?CD?C___gAO?G??KOcGCiA??SC????GAVQy????CQ?cCACKC_?A?E_??g_AO@C??c??@@?pY?G?")
6182
ce16.name(new = "ce16")
6183
add_to_lists(ce16, graph_objects, counter_examples, all_graphs)
6184
6185
# CE to independence_number(x) >= 1/2*cvetkovic(x)
6186
ce17 = Graph("S??wG@@h_GWC?AHG?_gMGY_FaIOk@?C?S")
6187
ce17.name(new = "ce17")
6188
add_to_lists(ce17, graph_objects, counter_examples, all_graphs)
6189
6190
# CE to independence_number(x) >= matching_number - sigma_dist2
6191
ce18 = Graph("cGO_?CCOB@O?oC?sTDSOCC@O???W??H?b???hO???A@CCKB??I??O??AO@CGA???CI?S?OGG?ACgQa_Cw^GP@AID?Gh??ogD_??dR[?AG?")
6192
ce18.name(new = "ce18")
6193
add_to_lists(ce18, graph_objects, counter_examples, all_graphs)
6194
6195
# CE to independence_number(x) <= maximum(max_even_minus_even_horizontal(x), radius(x)*welsh_powell(x))
6196
ce19 = Graph('J?@OOGCgO{_')
6197
ce19.name(new = "ce19")
6198
add_to_lists(ce19, graph_objects, counter_examples, all_graphs)
6199
6200
# CE to independence_number(x) <= card_center(x) + max_even_minus_even_horizontal(x) + 1
6201
ce20 = Graph('M?CO?k?OWEQO_O]c_')
6202
ce20.name(new = "ce20")
6203
add_to_lists(ce20, graph_objects, counter_examples, all_graphs)
6204
6205
# CE to independence_number(x) <= median_degree(x)^2 + card_periphery(x)
6206
ce21 = Graph('FiQ?_')
6207
ce21.name(new = "ce21")
6208
add_to_lists(ce21, graph_objects, counter_examples, all_graphs)
6209
6210
# CE to independence_number(x) <= brinkmann_steffen(x) + max_even_minus_even_horizontal(x) + 1
6211
ce22 = Graph('Ss?fB_DYUg?gokTEAHC@ECSMQI?OO?GD?')
6212
ce22.name(new = "ce22")
6213
add_to_lists(ce22, graph_objects, counter_examples, all_graphs)
6214
6215
# CE to independence_number(x) <= inverse_degree(x) + order_automorphism_group(x) + 1
6216
ce23 = Graph("HkIU|eA")
6217
ce23.name(new = "ce23")
6218
add_to_lists(ce23, graph_objects, counter_examples, all_graphs)
6219
6220
# CE to independence_number(x) <= ceil(eulerian_faces(x)/diameter(x)) +max_even_minus_even_horizontal(x)
6221
ce24 = Graph('JCbcA?@@AG?')
6222
ce24.name(new = "ce24")
6223
add_to_lists(ce24, graph_objects, counter_examples, all_graphs)
6224
6225
# CE to independence_number(x) <= floor(e^(maximum(max_even_minus_even_horizontal(x), fiedler(x))))
6226
ce25 = Graph('OX??ZHEDxLvId_rgaC@SA')
6227
ce25.name(new = "ce25")
6228
add_to_lists(ce25, graph_objects, counter_examples, all_graphs)
6229
6230
# CE to independence_number(x) <= maximum(card_periphery(x), radius(x)*welsh_powell(x))
6231
ce26 = Graph("NF?_?o@?Oa?BC_?OOaO")
6232
ce26.name(new = "ce26")
6233
add_to_lists(ce26, graph_objects, counter_examples, all_graphs)
6234
6235
# CE to independence_number(x) <= floor(average_distance(x)) + maximum(max_even_minus_even_horizontal(x), brinkmann_steffen(x))
6236
ce27 = Graph("K_GBXS`ysCE_")
6237
ce27.name(new = "ce27")
6238
add_to_lists(ce27, graph_objects, counter_examples, all_graphs)
6239
6240
# CE to independence_number(x) <= minimum(annihilation_number(x), 2*e^sum_temperatures(x))
6241
ce28 = Graph("g??O?C_?`?@?O??A?A????????C?????G?????????A@aA??_???G??GA?@????????_???GHC???CG?_???@??_??OB?C?_??????_???G???C?O?????O??A??????G??")
6242
ce28.name(new = "ce28")
6243
add_to_lists(ce28, graph_objects, counter_examples, all_graphs)
6244
6245
# CE to independence_number(x) <= maximum(2*welsh_powell(x), maximum(max_even_minus_even_horizontal(x), laplacian_energy(x)))
6246
ce29 = Graph("P@g??BSCcIA???COcSO@@O@c")
6247
ce29.name(new = "ce29")
6248
add_to_lists(ce29, graph_objects, counter_examples, all_graphs)
6249
6250
# CE to independence_number(x) <= maximum(order_automorphism_group(x), 2*cvetkovic(x) - matching_number(x))
6251
ce30 = Graph("G~q|{W")
6252
ce30.name(new = "ce30")
6253
add_to_lists(ce30, graph_objects, counter_examples, all_graphs)
6254
6255
# CE to independence_number(x) <= max_even_minus_even_horizontal(x) + min_degree(x) + welsh_powell(x)
6256
ce31 = Graph("VP??oq_?PDOGhAwS??bSS_nOo?OHBqPi?I@AGP?POAi?")
6257
ce31.name(new = "ce31")
6258
add_to_lists(ce31, graph_objects, counter_examples, all_graphs)
6259
6260
# CE to independence_number(x) >= order(x)/szekeres_wilf(x)
6261
ce32 = Graph('H?`@Cbg')
6262
ce32.name(new = "ce32")
6263
add_to_lists(ce32, graph_objects, counter_examples, all_graphs)
6264
6265
# CE to independence_number(x) <= max_even_minus_even_horizontal(x) + minimum(card_positive_eigenvalues(x), card_center(x) + 1)
6266
ce33 = Graph("O_aHgP_kVSGOCXAiODcA_")
6267
ce33.name(new = "ce33")
6268
add_to_lists(ce33, graph_objects, counter_examples, all_graphs)
6269
6270
# CE to independence_number(x) <= card_center(x) + maximum(diameter(x), card_periphery(x))
6271
ce34 = Graph('H?PA_F_')
6272
ce34.name(new = "ce34")
6273
add_to_lists(ce34, graph_objects, counter_examples, all_graphs)
6274
6275
# CE to independence_number(x) <= card_center(x) + maximum(diameter(x), card_periphery(x))ce35 = Graph("")
6276
ce35 = Graph("HD`cgGO")
6277
ce35.name(new = "ce35")
6278
add_to_lists(ce35, graph_objects, counter_examples, all_graphs)
6279
6280
# CE to independence_number(x) >= max_degree(x) - order_automorphism_group(x)
6281
ce36 = Graph('ETzw')
6282
ce36.name(new = "ce36")
6283
add_to_lists(ce36, graph_objects, counter_examples, all_graphs)
6284
6285
# CE to independence_number(x) <= maximum(card_center(x), diameter(x)*max_degree(x))
6286
ce37 = Graph("~?AA?G?????@@??@?A???????????O??????????G_?A???????????????A?AO?????????G???G?@???@???O?????????????C???????_???????C?_?W???C????????_??????????????????_???????_???O????????D??????????C????????GCC???A??G??????A@??A??@G???_?????@_??????_??G???K??????A????C??????????A???_?A????`??C_O????G????????????A?G???????????????????O?????C??????@???__?@O_G??C????????OA?????????????????????????GA_GA????O???_??O??O?G??G?_C???@?G???O???_?O???_??????C???????????????E_???????????????_@???O??????CC???O?????????OC_????_A????????_?G??????O??????_??????_?I?O??????A???????O?G?O???C@????????????_@????C?????@@???????C???O??A?????_??????A_??????????A?G????AB???A??C?G??????????G???A??@?A???????@???????D?_????B????????????????????g?C???C????G????????@??????@??A????????@????_??_???o?????????@????????????_???????A??????C????A?????C????O????@?@???@?A_????????CA????????????????H???????????????????O????_??OG??Ec?????O??A??_???_???O?C??`?_@??@??????O????G????????????A????@???_?????????_?A???AAG???O????????????????????C???_???@????????????_??H???A??W?O@????@_???O?_A??O????OG???????G?@??G?C?????G?????????@?????????G?O?????G???????_?????????@????@?????????G????????????C?G?????????_C?@?A????G??GA@????????????@?????C??G??????_?????????_@?????@???A?????@?????????????????CG??????_?????@???????@C???O????_`?????OA?G??????????????Q?A?????????????A????@C?????GO??_?C???????O???????@?G?A????O??G???_????_?????A?G_?C?????????C?")
6287
ce37.name(new = "ce37")
6288
add_to_lists(ce37, graph_objects, counter_examples, all_graphs)
6289
6290
# CE to independence_number(x) <= abs(-card_center(x) + min_degree(x)) + max_even_minus_even_horizontal(x)
6291
ce38 = Graph('FVS_O')
6292
ce38.name(new = "ce38")
6293
add_to_lists(ce38, graph_objects, counter_examples, all_graphs)
6294
6295
# CE to independence_number(x) <= abs(-card_center(x) + max_degree(x)) + max_even_minus_even_horizontal(x)
6296
ce39 = Graph("FBAuo")
6297
ce39.name(new = "ce39")
6298
add_to_lists(ce39, graph_objects, counter_examples, all_graphs)
6299
6300
# CE to independence_number(x) <= floor(inverse_degree(x)) + order_automorphism_group(x) + 1
6301
ce40 = Graph('Htji~Ei')
6302
ce40.name(new = "ce40")
6303
add_to_lists(ce40, graph_objects, counter_examples, all_graphs)
6304
6305
# CE to independence_number(x) <= card_center(x) + maximum(residue(x), card_periphery(x))
6306
ce42 = Graph('GP[KGC')
6307
ce42.name(new = "ce42")
6308
add_to_lists(ce42, graph_objects, counter_examples, all_graphs)
6309
6310
# CE to independence_number(x) <= maximum(girth(x), (barrus_bound(x) - order_automorphism_group(x))^2)
6311
ce43 = Graph("Exi?")
6312
ce43.name(new = "ce43")
6313
add_to_lists(ce43, graph_objects, counter_examples, all_graphs)
6314
6315
# CE to independence_number(x) <= (brinkmann_steffen(x) - szekeres_wilf(x))^2 + max_even_minus_even_horizontal(x)
6316
ce44 = Graph('GGDSsg')
6317
ce44.name(new = "ce44")
6318
add_to_lists(ce44, graph_objects, counter_examples, all_graphs)
6319
6320
# CE to independence_number(x) <= maximum(max_even_minus_even_horizontal(x), radius(x)*szekeres_wilf(x))
6321
ce45 = Graph("FWKH?")
6322
ce45.name(new = "ce45")
6323
add_to_lists(ce45, graph_objects, counter_examples, all_graphs)
6324
6325
# CE to independence_number(x) <= maximum(card_periphery(x), radius(x)*szekeres_wilf(x))
6326
ce46 = Graph('F`I`?')
6327
ce46.name(new = "ce46")
6328
add_to_lists(ce46, graph_objects, counter_examples, all_graphs)
6329
6330
# CE to independence_number(x) <= maximum(card_periphery(x), diameter(x) + inverse_degree(x))
6331
ce47 = Graph("KVOzWAxewcaE")
6332
ce47.name(new = "ce47")
6333
add_to_lists(ce47, graph_objects, counter_examples, all_graphs)
6334
6335
# CE to independence_number(x) <= maximum(card_periphery(x), max_even_minus_even_horizontal(x) + min_degree(x))
6336
ce48 = Graph('Iq]ED@_s?')
6337
ce48.name(new = "ce48")
6338
add_to_lists(ce48, graph_objects, counter_examples, all_graphs)
6339
6340
# CE to independence_number(x) >= sqrt(card_positive_eigenvalues(x))
6341
ce49 = Graph("K^~lmrvv{~~Z")
6342
ce49.name(new = "ce49")
6343
add_to_lists(ce49, graph_objects, counter_examples, all_graphs)
6344
6345
# CE to independence_number(x) <= max_degree(x) + maximum(max_even_minus_even_horizontal(x), sigma_dist2(x))
6346
ce50 = Graph('bCaJf?A_??GY_O?KEGA???OMP@PG???G?CO@OOWO@@m?a?WPWI?G_A_?C`OIG?EDAIQ?PG???A_A?C??CC@_G?GDI]CYG??GA_A??')
6347
ce50.name(new = "ce50")
6348
add_to_lists(ce50, graph_objects, counter_examples, all_graphs)
6349
6350
# CE to independence_number(x) >= matching_number(x) - order_automorphism_group(x) - 1
6351
ce51 = Graph("Ivq~j^~vw")
6352
ce51.name(new = "ce51")
6353
add_to_lists(ce51, graph_objects, counter_examples, all_graphs)
6354
6355
# CE to independence_number(x) >= order(x)/szekeres_wilf(x)
6356
ce52 = Graph('H?QaOiG')
6357
ce52.name(new = "ce52")
6358
add_to_lists(ce52, graph_objects, counter_examples, all_graphs)
6359
6360
# CE to independence_number(x) >= matching_number(x) - sigma_dist2(x) - 1
6361
ce53 = Graph("]?GEPCGg]S?`@??_EM@OTp?@E_gm?GW_og?pWO?_??GQ?A?^HIRwH?Y?__BC?G?[PD@Gs[O?GW")
6362
ce53.name(new = "ce53")
6363
add_to_lists(ce53, graph_objects, counter_examples, all_graphs)
6364
6365
# CE to independence_number(x) >= -average_distance(x) + ceil(lovasz_theta(x))
6366
ce54 = Graph('lckMIWzcWDsSQ_xTlFX?AoCbEC?f^xwGHOA_q?m`PDDvicEWP`qA@``?OEySJX_SQHPc_H@RMGiM}`CiG?HCsm_JO?QhI`?ARLAcdBAaOh_QMG?`D_o_FvQgHGHD?sKLEAR^ASOW~uAUQcA?SoD?_@wECSKEc?GCX@`DkC')
6367
ce54.name(new = "ce54")
6368
add_to_lists(ce54, graph_objects, counter_examples, all_graphs)
6369
6370
# CE to independence_number(x) >= -card_periphery(x) + matching_number(x)
6371
ce55 = Graph("I~~~~~~zw")
6372
ce55.name(new = "ce55")
6373
add_to_lists(ce55, graph_objects, counter_examples, all_graphs)
6374
6375
# CE to independence_number(x) >= lovasz_theta(x)/edge_con(x)
6376
ce56 = Graph('HsaGpOe')
6377
ce56.name(new = "ce56")
6378
add_to_lists(ce56, graph_objects, counter_examples, all_graphs)
6379
6380
# CE to independence_number(x) >= minimum(max_degree(x), floor(lovasz_theta(x)))
6381
ce57 = Graph("^?H{BDHqHosG??OkHOhE??B[CInU?@j_A?CoA^azGPLcb_@GEYYRPgG?K@gdPAg?d@_?_sGcED`@``O")
6382
ce57.name(new = "ce57")
6383
add_to_lists(ce57, graph_objects, counter_examples, all_graphs)
6384
6385
# CE to independence_number>= barrus_bound(x) - max(card_center(x), card_positive_eigenvalues(x))
6386
ce58 = Graph('Sj[{Eb~on~nls~NJWLVz~~^|{l]b\uFss')
6387
ce58.name(new = "ce58")
6388
add_to_lists(ce58, graph_objects, counter_examples, all_graphs)
6389
6390
# CE to independence_number(x) >= floor(tan(barrus_bound(x) - 1))
6391
ce59 = Graph("RxCWGCB?G?_B?@??_?N??F??B_??w?")
6392
ce59.name(new = "ce59")
6393
add_to_lists(ce59, graph_objects, counter_examples, all_graphs)
6394
6395
# CE to independence_number(x) >= -1/2*diameter(x) + lovasz_theta(x)
6396
ce60 = Graph('wSh[?GCfclJm?hmgA^We?Q_KIXbf\@SgDNxpwHTQIsIB?MIDZukArBAeXE`vqDLbHCwf{fD?bKSVLklQHspD`Lo@cQlEBFSheAH?yW\YOCeaqmOfsZ?rmOSM?}HwPCIAYLdFx?o[B?]ZYb~IK~Z`ol~Ux[B]tYUE`_gnVyHRQ?{cXG?k\BL?vVGGtCufY@JIQYjByg?Q?Qb`SKM`@[BVCKDcMxF|ADGGMBW`ANV_IKw??DRkY\KOCW??P_?ExJDSAg')
6397
ce60.name(new = "ce60")
6398
add_to_lists(ce60, graph_objects, counter_examples, all_graphs)
6399
6400
# CE to independence_number(x) <= maximum(card_negative_eigenvalues(x), max_common_neighbors(x) + max_even_minus_even_horizontal(x))
6401
ce61 = Graph("KsaAA?OOC??C")
6402
ce61.name(new = "ce61")
6403
add_to_lists(ce61, graph_objects, counter_examples, all_graphs)
6404
6405
# CE to independence_number(x) >= minimum(floor(lovasz_theta(x)), tan(spanning_trees_count(x)))
6406
ce62 = Graph("qWGh???BLQcAH`aBAGCScC@SoBAAFYAG?_T@@WOEBgRC`oSE`SG@IoRCK[_K@QaQq?c@?__G}ScHO{EcCa?K?o?E?@?C[F_@GpV?K_?_?CSW@D_OCr?b_XOag??C@gGOGh??QFoS?@OHDAKWIX_OBbHGOl??\Cb@?E`WehiP@IGAFC`GaCgC?JjQ???AGJgDJAGsdcqEA_a_q?")
6407
ce62.name(new = "ce62")
6408
add_to_lists(ce62, graph_objects, counter_examples, all_graphs)
6409
6410
# CE to independence_number(x) >= diameter(x)/different_degrees(x)
6411
ce63 = Graph("KOGkYBOCOAi@")
6412
ce63.name(new = "ce63")
6413
add_to_lists(ce63, graph_objects, counter_examples, all_graphs)
6414
6415
# CE to independence_number(x) >= -max_common_neighbors(x) + min_degree(x)
6416
ce64 = Graph('`szvym|h~RMQLTNNiZzsgQynDR\p~~rTZXi~n`kVvKolVJfP}TVEN}Thj~tv^KJ}D~VqqsNy|NY|ybklZLnz~TfyG')
6417
ce64.name(new = "ce64")
6418
add_to_lists(ce64, graph_objects, counter_examples, all_graphs)
6419
6420
# CE to independence_number(x) >= -10^different_degrees(x) + matching_number(x)
6421
ce65 = Graph("W~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~")
6422
ce65.name(new = "ce65")
6423
add_to_lists(ce65, graph_objects, counter_examples, all_graphs)
6424
6425
# CE to independence_number(x) >= girth^max_degree+1
6426
ce66 = Graph("~?@EG??????????@G????_???a???C????????@???A???????G??????C?GCG????????A???C@??????@????O??A??C?????_??O???CA???c??_?_?@????A????@??????C???C?G?O?C???G?????????O?_G?C????G??????_?????@??G???C??????O?GA?????O???@????????A?G?????????_C???????@??G??@??_??IA@???????G?@??????@??_?@????C??G???_????O???P???@???o??????O?????S?O???A???G?????c_?????D?????A???A?????G@???????O???H????O????@@????@K????????C??C?????G??")
6427
ce66.name(new = "ce66")
6428
add_to_lists(ce66, graph_objects, counter_examples, all_graphs)
6429
6430
# CE to independence_number(x) <= maximum(cycle_space_dimension(x), floor(lovasz_theta(x)))
6431
ce67 = Graph("G??EDw")
6432
ce67.name(new = "ce67")
6433
add_to_lists(ce67, graph_objects, counter_examples, all_graphs)
6434
6435
# CE to independence_number(x) >= minimum(card_positive_eigenvalues(x), 2*card_zero_eigenvalues(x))
6436
ce68 = Graph('HzzP|~]')
6437
ce68.name(new = "ce68")
6438
add_to_lists(ce68, graph_objects, counter_examples, all_graphs)
6439
6440
# CE to independence_number(x) <= maximum(max_degree(x), radius(x)^card_periphery(x))
6441
ce69 = Graph("F?BvO")
6442
ce69.name(new = "ce69")
6443
add_to_lists(ce69, graph_objects, counter_examples, all_graphs)
6444
6445
# CE to independence_number(x) >= floor(lovasz_theta(x))/vertex_con(x)
6446
ce70 = Graph('~?@Z??????O?M??`S??A?`?A?????@????`?????A?A?????A@????GO?@@??A_????????O_???I@_??G??A?`?C????????@???????????@??C?@?????O??@??CA??A?D??G?_?????_Q@G????C?_?A??@???O????G?O?G?_?????CoG?G???X??C???_CAG_C??????G?????@?Ao?????C???A??????_??SG??cOC??????????Ao????????_?????G???????D?????C??_?B?????a??_???????G?@?????C??????C?c?????G_?_??G??_Q????C????B?_CG????AGC???G?O??_I????@??????_??a??@?O_G??O??aA@@?????EA???@???????@???????O?O??@??`_G???????GCA?_GO????_?_????????????_??I?@?C???@????????G?aG??????W????@PO@???oC?CO???_??G?@@?CO??K???C@??O???@????D?????A?@G?G?O???_???????Ao??AC???G?_???G????????A??????_?p???W?A?Ao@?????_?????GA??????????????_?C??????@O????_@??O@Gc@??????????A_??????')
6447
ce70.name(new = "ce70")
6448
add_to_lists(ce70, graph_objects, counter_examples, all_graphs)
6449
6450
# CE to independence_number(x) <= maximum(matching_number(x), critical_independence_number(x))
6451
ce71 = Graph('ECYW')
6452
ce71.name(new = "ce71")
6453
add_to_lists(ce71, graph_objects, counter_examples, all_graphs)
6454
6455
# CE to independence_number(x)>=-1/2*x.diameter() + x.lovasz_theta()
6456
ce72 = Graph('fdSYkICGVs_m_TPs`Fmj_|pGhC@@_[@xWawsgEDe_@g`TC{P@pqGoocqOw?HBDS[R?CdG\e@kMCcgqr?G`NHGXgYpVGCoJdOKBJQAsG|ICE_BeMQGOwKqSd\W?CRg')
6457
ce72.name(new = "ce72")
6458
add_to_lists(ce72, graph_objects, counter_examples, all_graphs)
6459
6460
# CE to independence_number(x) >= minimum(floor(lovasz_theta(x)), max_even_minus_even_horizontal(x) + 1)
6461
ce73 = Graph('h???_?CA?A?@AA????OPGoC@????A@?A?_C?C?C_A_???_??_G????HG????c?G_?G??HC??A@GO?G?A@A???_@G_?_G_GC_??E?O?O`??@C?@???O@?AOC?G?H??O?P??C_?O_@??')
6462
ce73.name(new = "ce73")
6463
add_to_lists(ce73, graph_objects, counter_examples, all_graphs)
6464
6465
# CE to independence_number(x) >= minimum(diameter(x), lovasz_theta(x))
6466
ce74 = Graph("FCQb_")
6467
ce74.name(new = "ce74")
6468
add_to_lists(ce74, graph_objects, counter_examples, all_graphs)
6469
6470
# CE to independence_number(x) >= minimum(girth(x), floor(lovasz_theta(x)))
6471
ce75 = Graph('E?Bw')
6472
ce75.name(new = "ce75")
6473
add_to_lists(ce75, graph_objects, counter_examples, all_graphs)
6474
6475
# CE to independence_number(x) <= maximum(average_distance(x), max_even_minus_even_horizontal(x))*sum_temperatures(x)
6476
ce76 = Graph("~?@DS?G???G_?A_?OA?GC??oa?A@?@?K???L?_?S_??CCSA_g???@D?????_?A??EO??GAOO_@C`???O?_CK_???_o_?@O??XA???AS???oE`?A?@?CAa?????C?G??i???C@qo?G?Og?_O?_?@???_G????o?A_@_?O?@??EcA???__?@GgO?O@oG?C?@??CIO?_??G??S?A?@oG_K?@C??@??QOA?C????AOo?p?G???oACAOAC@???OG??qC???C??AC_G?@??GCHG?AC@?_@O?CK?@?B???AI??OO_S_a_O??????AO?OHG?@?????_???EGOG??@?EF@?C?Pc?????C?W_PA?O@?_?@A@??OD_C?@?@?A??CC?_?i@?K?_O_CG??A?")
6477
ce76.name(new = "ce76")
6478
add_to_lists(ce76, graph_objects, counter_examples, all_graphs)
6479
6480
# CE to independence_number(x) <= maximum(matching_number(x), critical_independence_number(x))
6481
ce77 = Graph("iF\ZccMAoW`Po_E_?qCP?Ag?OGGOGOS?GOH??oAAS??@CG?AA?@@_??_P??G?SO?AGA??M????SA????I?G?I???Oe?????OO???_S?A??A????ECA??C?A@??O??S?@????_@?_??S???O??")
6482
ce77.name(new = "ce77")
6483
add_to_lists(ce77, graph_objects, counter_examples, all_graphs)
6484
6485
# CE to independence_number(x) <= maximum(max_degree(x), radius(x)^card_periphery(x))
6486
ce78 = Graph("G_aCp[")
6487
ce78.name(new = "ce78")
6488
add_to_lists(ce78, graph_objects, counter_examples, all_graphs)
6489
6490
# CE to independence_number(x) <= residue(x)^2
6491
ce79 = Graph('J?B|~fpwsw_')
6492
ce79.name(new = "ce79")
6493
add_to_lists(ce79, graph_objects, counter_examples, all_graphs)
6494
6495
# CE to independence_number(x) <= 10^(card_center(x)*log(10)/log(sigma_dist2(x)))
6496
ce80 = Graph('T?????????????????F~~~v}~|zn}ztn}zt^')
6497
ce80.name(new = "ce80")
6498
add_to_lists(ce80, graph_objects, counter_examples, all_graphs)
6499
6500
# CE to independence_number(x) <= diameter(x)^card_periphery(x)
6501
ce81 = Graph('P?????????^~v~V~rzyZ~du{')
6502
ce81.name(new = "ce81")
6503
add_to_lists(ce81, graph_objects, counter_examples, all_graphs)
6504
6505
# CE to independence_number(x) <= radius(x)*residue(x) + girth(x)
6506
ce82 = Graph('O????B~~^Zx^wnc~ENqxY')
6507
ce82.name(new = "ce82")
6508
add_to_lists(ce82, graph_objects, counter_examples, all_graphs)
6509
6510
"""
6511
CE to independence_number(x) <= minimum(lovasz_theta(x), residue(x)^2)
6512
and
6513
<= minimum(annihilation_number(x), residue(x)^2)
6514
and
6515
<= minimum(fractional_alpha(x), residue(x)^2)
6516
and
6517
<= minimum(cvetkovic(x), residue(x)^2)
6518
and
6519
<= minimum(residue(x)^2, floor(lovasz_theta(x)))
6520
and
6521
<= minimum(size(x), residue(x)^2)
6522
"""
6523
ce83 = Graph('LEYSrG|mrQ[ppi')
6524
ce83.name(new = "ce83")
6525
add_to_lists(ce83, graph_objects, counter_examples, all_graphs)
6526
6527
# CE to independence_number(x) <= maximum(laplacian_energy(x), brinkmann_steffen(x)^2)
6528
ce84 = Graph('~?@r?A??OA?C??????@?A????CC?_?A@????A?@???@?S?O????AO??????G???????C????????C?C???G?????_??????_?G?????O?A?_?O?O@??O???T@@??????O????C_???C?CO???@??@?@???_???O??O??A??O???O?A?OB?C?AD???C`?B?__?_????????Q?C??????????????_???C??_???A?gO??@C???C?EC?O??GG`?O?_?_??O????_?@?GA?_????????????G????????????????????AO_?C?????????P?IO??I??OC???O????A??AC@AO?o????????o@??O?aI?????????_A??O??G??o?????????_??@?????A?O?O?????G?????H???_????????A??a?O@O?_?D???????O@?????G???GG?CA??@?A@?A????GA?@???G??O??A??????AA???????O??_c??@???A?????_????@CG????????????A???A???????A?W???B????@?????HGO???????_@_?????C??????????_a??????_???????@G?@O?@@_??G@???????GG?O??A??????@????_??O_?_??CC?B???O??@????W??`AA????O??_?????????????????_???A??????@G??????I@C?G????????A@?@@?????C???p???????????????????G?_G????Z?A????_??????G????Q????@????????_@O????@???_QC?A??@???o???G???@???????O???CC??O?D?O?@C????@O?G?????A??@C???@????O?????????_??C??????_?@????O??????O?Y?C???_?????A??@OoG???????A???G??????CC??A?A?????????????????GA_???o???G??O??C???_@@??????@?????G??????????O???@O???????????A????S??_o????????A??B??????_??C????C?')
6529
ce84.name(new = "ce84")
6530
add_to_lists(ce84, graph_objects, counter_examples, all_graphs)
6531
6532
# CE to independence_number(x) <= girth(x)^ceil(laplacian_energy(x))
6533
ce85 = Graph('bd_OPG_J_G?apBB?CPk@`X?hB_?QKEo_op`C?|Gc?K_?P@GCoGPTcGCh?CBIlqf_GQ]C_?@jlFP?KSEALWGi?bIS?PjO@?CCA?OG?')
6534
ce85.name(new = "ce85")
6535
add_to_lists(ce85, graph_objects, counter_examples, all_graphs)
6536
6537
# CE to independence_number(x) <= diameter(x)*residue(x) + different_degrees(x)
6538
ce86 = Graph('SK|KWYc|^BJKlaCnMH^ECUoSC[{LHxfMG')
6539
ce86.name(new = "ce86")
6540
add_to_lists(ce86, graph_objects, counter_examples, all_graphs)
6541
6542
# CE to independence_number(x) <= maximum(max_common_neighbors(x), girth(x)^laplacian_energy(x))
6543
ce87 = Graph('~?@iA?B@@b?[a??oHh_gC?@AGD?Wa???_E@_@o_AkGA@_o?_h??GG??cO??g?_?SD?d@IW?s?P_@@cSG?_B??d?CSI?OCQOH_?bo?CAaC???pGC@DO@?PHQOpO??A?_A@K[PC@?__???@OSCOGLO?oOAAA?IOX@??GC?O?P??oA_?KPIK?Q@A?sQC???LA???aQOC_AeG?Q?K_Oo?AB?OU?COD?VoQ?@D????A?_D?CAa?@@G?C??CGHcCA_cB?@c@_O?H??_@?@OWGGCo??AGC??AQ?QOc???Ow_?C[?O@@G_QH?H?O???_I@@PO????FAGk??C?ka@D@I?P?CooC@_O@?agAE??CpG?AA_`OO??_?Q?AiOQEK?GhB@CAOG?G?CC??C@O@GdC__?OIBKO?aOD_?OG???GACH@?b?@?B_???WPA?@_?o?XQQ?ZI_@?O_o_?@O??EDGOBEA??_aOSQsCO@?_DD`O??D?JaoP?G?AOQOCAS?k??S?c@?XW?QCO??_OAGOWc__G?_??G??L@OP?b?O?GCCMAH????????@@?A?C@oDaGG?Wk@H@OM?_A?IOu`SG?E@??W?I@EQA@@_@Wa?@?_??C??AAAiGQG@@?`@oA?_??OgC?K_G??G`?@S@B?A?HWc?HG??`gO???A?W?A?O?MpS??D?GS?GDC_??I@??IPAOdk`?CG??A?pPAgIDlCYCTSDgg?@FW?DI?O_OW?_S??AAQB_OOCF????XS_?@l_kAw__Ea?O?C_CGO??EG??WLb@_H??OCaAET@S?@?I???_??LaO_HCYG@G_G?_?_C???os?_G?OO@s_??_?_GGE`Os??_GCa?DWO?A@?@_CB`MOBCGIC???GKA_c?@BSh??@?RC[?eg?@hOC?_?BeGOaC?AWOSCm@G?A??A?G?Ga_')
6544
ce87.name(new = "ce87")
6545
add_to_lists(ce87, graph_objects, counter_examples, all_graphs)
6546
6547
# CE to independence_number(x) <= maximum(radius(x), max_degree(x))^2
6548
ce88 = Graph('h@`CA???GH?AAG?OW@@????E???O?O???PO?O?_?G??`?O_???@??E?E??O??A?S@???S???????U?GAI???A?DA??C?C@??PA?A???_C_?H?AA??_C??DCO?C???_?AAG??@O?_?G')
6549
ce88.name(new = "ce88")
6550
add_to_lists(ce88, graph_objects, counter_examples, all_graphs)
6551
6552
# CE to independence_number(x) <= max_degree(x) + maximum(max_even_minus_even_horizontal(x), geometric_length_of_degree_sequence(x))
6553
ce89 = Graph("_qH?S@??`??GG??O?_?C?_??@??@??G??C??_??C????O??G???@????O???A???@????C???C?????G????")
6554
ce89.name(new = "ce89")
6555
add_to_lists(ce89, graph_objects, counter_examples, all_graphs)
6556
6557
# CE to independence_number(x) >= floor(arccosh(lovasz_theta(x)))^2
6558
ce90 = Graph("~?@Td|wi\\fbna~}wepkkbXcrW}\\~NvtLKpY\\J_Ub^~yM~^tHnM}jPffKkqnijvxD@xa{UOzzvr?L^PFi||yt@OQ\YU{Vh]tWzwzpj\\n|kR]`Y}RpCvxk{rEMRP\\}|}dNdNtbO~yrkgMxlOXr|FvQ{tvfKKnHrp^}jV\\B^n\\LvLZeyX}QSKN^sm~yl\\[NJZXqdk]O|^zHl~vC{w`Nsn}x]utqrJozKXV|eIUUPv~ydc}]xJNWZjW|lpYm}{Jf~JWMixb^t]e|S~B[vKc{K[Kjut~}Kj~iAl\\tVNgyZadvoA}rdTlr\\\\wNr^^kJzrp|qlVy]siKncI~`oNm|ul\\PxDRyzddDzrjUn~ciOgbR}p~Cz|~MlxYoEVnVuZkxJgvmtE]]}~PRp[He]oBQz]PVJ~gVnvSUR|QF|`lomFh[j|jIaS~vh~_rYiiK}FnEW}ovnntxtRFBakzvwn[biJhNvf|VDV?m~Y]ndmfJQ|M@QvnNf~MCyn~{HSU~fvEv~@}u|spOXzTVNY\\kjDNt\\zRMXxU|g|XrzFzDYiVvho}bQbyfI{{w[_~nrm}J~LhwH}TNmfM^}jqajl_ChY]M}unRK\\~ku")
6559
ce90.name(new = "ce90")
6560
add_to_lists(ce90, graph_objects, counter_examples, all_graphs)
6561
6562
# CE to independence_number(x) <= maximum(2*welsh_powell(x), max_even_minus_even_horizontal(x)^2)
6563
ce91 = Graph("q?}x{k\FGNCRacDO`_gWKAq?ED?Qc?IS?Da?@_E?WO_@GOG@B@?Cc?@@OW???qO?@CC@?CA@C?E@?O?KK???E??GC?CO?CGGI??@?cGO??HG??@??G?SC???AGCO?KAG???@O_O???K?GG????WCG??C?C??_C????q??@D??AO???S????CA?a??A?G??IOO????B?A???_??")
6564
ce91.name(new = "ce91")
6565
add_to_lists(ce91, graph_objects, counter_examples, all_graphs)
6566
6567
# CE to independence_number(x) >= -max_common_neighbors(x) + min_degree(x) - 1
6568
ce92 = Graph("qR}fexr{J\\innanomndYrzmy^p~Ri]c]lA{~jVurv]n~reCed~|j{TtvnMtB~nZFrz{wUnV^fzV\\rUlt|qvJubnFwWSxxzfZ}Btj`yV~rv\\nknwl~Z?T]{qwn~bFzh^\\{Ezv}p~I^RV|oXe~knL~x^nNtvYlrezLX^tj{S^Rflqqv]e|S^}vpbe~Ni]m]}zfbZolnPl{N~}]X?")
6569
ce92.name(new = "ce92")
6570
add_to_lists(ce92, graph_objects, counter_examples, all_graphs)
6571
6572
# CE to independence_number(x) >= diameter for regular graphs
6573
# Ryan Pepper, June 2017
6574
ce93 = Graph('_t???CA???_B?E?@??WAB?G??_GA?W?????@???W??B???A???BA??@__???G??@A???LA???AW???@_???G')
6575
ce93.name(new = "ce93")
6576
add_to_lists(ce93, graph_objects, counter_examples, all_graphs)
6577
6578
# CE to independence_number(x) >= diameter for regular graphs
6579
# Ryan Pepper, June 2017
6580
ce94 = Graph('Yv?GW?@?WB?A?A?@_?oGA?_KG????G??K??A???W??@???AO??BG??B?')
6581
ce94.name(new = "ce94")
6582
add_to_lists(ce94, graph_objects, counter_examples, all_graphs)
6583
6584
# CE to independence_number(x) >= -average_distance(x) + ceil(lovasz_theta(x))
6585
ce95 = Graph('epCpih@K}gSGIZfc?Rkf{EWtVKJTmJtWYl_IoDOKOikwDSKtbH\\fi_g`affO\\|Agq`WcLoakSLNPaWZ@PQhCh?ylTR\\tIR?WfoJNYJ@B{GiOWMUZX_puFP?')
6586
ce95.name(new = "ce95")
6587
add_to_lists(ce95, graph_objects, counter_examples, all_graphs)
6588
6589
# CE to independence_number(x) >= ceil(1/2*cvetkovic(x))
6590
# CE to independence_number(x) >= 1/2*cvetkovic(x)
6591
ce96 = Graph('Gvz~r{')
6592
ce96.name(new = "ce96")
6593
add_to_lists(ce96, graph_objects, counter_examples, all_graphs)
6594
6595
# CE to independence_number(x) >= -max_common_neighbors(x) + min_degree(x)
6596
ce97 = Graph('eLvnv~yv{yJ~rlB^Mn|v^nz~V]mwVji}^vZf{\\\\nqZLfVBze}y[ym|jevvt~NNucret|~ejj~rz}Q\\~^an}XzTV~t]a]v}nx\\u]n{}~ffqjn`~e}lZvV]t_')
6597
ce97.name(new = "ce97")
6598
add_to_lists(ce97, graph_objects, counter_examples, all_graphs)
6599
6600
# CE to independence_number(x) >= lovasz_theta(x)/radius(x)
6601
ce100 = Graph('~?@A~~~~~~z~~~~~~~~^~~~~~z~~~~~~~~~~~~~~~v~~~v~~~~~~~~~~z~~~~~~~~^~~~~~~~~~}\~~}v~^~~}~~^~~~~~~~~~~~~^~~~~~~~~V~~~n~~n~~~~~~}~~|~}~~~~~~~~~~~~~~~~~~~~~vv~|~~~~~~~~~~~~~~~~~z~~w~~~~~~~~~~~~~~~~n~~~|~~~~~~~v~|~~~~~~~~~~}~|~r~V~~~n~~~~~~~~z~~}}~}~~~~vz~~z~~~z}~~~n~~~~~~~~~~~~n~~~~~~~z~~~~~~~~~~~~~~^~~~~~~~~~n~~]~~~~~n~~~}~~~~~~~~~~^~^~~~~}~~~~~~~~~~~z~~~~^~~~~~~w')
6602
ce100.name(new = "ce100")
6603
add_to_lists(ce100, graph_objects, counter_examples, all_graphs)
6604
6605
# CE to independence_number(x) >= matching_number(x) - order_automorphism_group(x) - 1
6606
ce101 = Graph('I~~Lt~\Nw')
6607
ce101.name(new = "ce101")
6608
add_to_lists(ce101, graph_objects, counter_examples, all_graphs)
6609
6610
# CE to independence_number(x) >= card_positive_eigenvalues(x) - lovasz_theta(x)
6611
ce102 = Graph('N^nN~~}Z~|}~~\~]zzw')
6612
ce102.name(new = "ce102")
6613
add_to_lists(ce102, graph_objects, counter_examples, all_graphs)
6614
6615
# CE to independence_number(x) >= card_negative_eigenvalues(x) - sigma_dist2(x)
6616
ce103 = Graph('IOilnemjG')
6617
ce103.name(new = "ce103")
6618
add_to_lists(ce103, graph_objects, counter_examples, all_graphs)
6619
6620
# CE to independence_number(x) >= minimum(min_degree(x), floor(lovasz_theta(x)))
6621
ce105 = Graph('z@M@E?OYOSCPBTp?mOWaP_?W[OG[abE_?P[@?@REt?ggAAOH?N@?CATE\oE?WO@GOKu?LJ_??SDP@CIA?AFHCC?kZQMo@CkOGoiJSs`?g?oDJqC?S?qJSqA?GN]?OPd?cGHE?AOpE_c_O@kC_?DF@HGgJ?ygAACdcCMPA[d`SHE@?PqRE?CO_?CWO?H_b_EBoOKI@CWAadQ?eOO?oT_@STAWCCCMOK?A@?TsBoJa@?PGQ_CiKPC_@_iE_hL@ACAIJQDgOSG?G[cG_D[A_CbDKO[goBH_?S?')
6622
ce105.name(new = "ce105")
6623
add_to_lists(ce105, graph_objects, counter_examples, all_graphs)
6624
6625
# CE to independence_number(x) <= minimum(floor(lovasz_theta(x)), 10^laplacian_energy(x))
6626
ce106 = Graph("k??????????????????????????????????????F~~~z~~~\~~~~{~^|nvny\^~~Njj~~zo}^yB|z~}h|Vv\Jft]~RlhV~tZMC^~fpvBynshVa~yw~@Tv\IVaJ}tvA\erD|Xx_rijkiIxLx}GE\pZ{yIwW?vV}K")
6627
ce106.name(new = "ce106")
6628
add_to_lists(ce106, graph_objects, counter_examples, all_graphs)
6629
6630
# CE to independence_number(x) <= max_even_minus_even_horizontal(x) + sinh(max_degree(x) - 1)
6631
ce107 = Graph("~?@S???CO@_??C??????O?D????A?@?O?_????????????_???__?_O???????G??????A@????C????????????H???G??????_?G???????@????????OG?G_?C????C??K????A?_O???O?O????O?O????OOG?G??????C???????g???????C@???????A?OG_???CO?_?_??CC????@?G?????OO?@?C??????_?????O???A???????O?i??????????O?C???O????G??@??A?G??????A???_??A????OC?G???C?@?O???G??A?O???G?A??????C??C???@??????A??@????O???@??G?_?_???_???@G@???@???G??_?C????A??A?@AA????E????G?????????@O??@A????A????@?G?????????@@??_C???????G??@?@_???????@?????K?_???GO????_???O?C?C?@G??????????CG??@?G??????CQ??????G???????C__??C???_A?????????A_??o????_?@????")
6632
ce107.name(new = "ce107")
6633
add_to_lists(ce107, graph_objects, counter_examples, all_graphs)
6634
6635
# CE to independence_number(x) <= 10^cosh(log(residue(x)) - 1)
6636
ce108 = Graph("U??????g}yRozOzw\wBn?zoBv_FN?Bn?B|??vO??")
6637
ce108.name(new = "ce108")
6638
add_to_lists(ce108, graph_objects, counter_examples, all_graphs)
6639
6640
# CE to independence_number(x) >= floor(caro_wei(x))^min_common_neighbors(x)
6641
ce109 = Graph("Z???????????????????????????????B~~~v~~z~~|~nvnnjz~~xfntuV~w")
6642
ce109.name(new = "ce109")
6643
add_to_lists(ce109, graph_objects, counter_examples, all_graphs)
6644
6645
# CE to independence_number(x) >= min(card_positive_eigenvalues(x), 2*card_zero_eigenvalues(x))
6646
ce110 = Graph("GUxvuw")
6647
ce110.name(new = "ce110")
6648
add_to_lists(ce110, graph_objects, counter_examples, all_graphs)
6649
6650
# CE to independence_number(x) >= minimum(lovasz_theta(x), max_degree(x)/card_center(x))
6651
ce112 = Graph("flthktLhme|L|L]efhg|LbstFhhFhgbssG|L`FhiC]ecG|LCG|LaC]ew`FhnCG|L{Obstw`Fhlw`Fhi{ObssnCG|Ldw`FhmVaC]eknCG|LknCG|L??????_?????G")
6652
ce112.name(new = "ce112")
6653
add_to_lists(ce112, graph_objects, counter_examples, all_graphs)
6654
6655
# CE to independence_number(x) <= maximum(e^average_distance(x), max_degree(x)^2)
6656
ce114 = Graph("~?@W?g??????????????????_???O??A???@c???W?S?P????O?@?@??G??????A????????O?????????????G??_?O?@G??????a??????GO???O????????C????O?A?@?AO???G_?AAG_?????o???????????????I??DG????????G??A_????C??O?G??_?_?????_G???????`??@??????A??C????OA?BG???????C?_???????a?@??O??_G?????????@@???O??????C@???A?A??G????????????@???_?????G?@O??????E???????????????O???@?E?O????G????C????A@????????A?????A_?Q???????AOO?@_?@??????????OG????g?????_?CG?????AA???O?G?????????__???????@??A??Q?????GOG??C????@??CC???????@???????G?C@???A??_?????A?????_????AC??C?@??????_?C??A???G???A???OGA????????A????_????K??O??????????@?????a???_??O??`??????A?????G???@?OOC??C???_?????")
6657
ce114.name(new = "ce114")
6658
add_to_lists(ce114, graph_objects, counter_examples, all_graphs)
6659
6660
# CE to independence_number(x) <= 2*wilf(x)^2 + radius(x)
6661
ce115 = Graph("z??C????????A????AA?_??G????????????C?????_??@?????????????G?@??C?????A??C?????C??????C?????????C??????????O?_?????????????G@??A?C???O?????AC?????????O???@???_?S??????I???C???O?G????????G??C?_????A_??CG?A?????????_????A????????????????G???C???????????CC?G???C?????_????????o?I???_???????")
6662
ce115.name(new = "ce115")
6663
add_to_lists(ce115, graph_objects, counter_examples, all_graphs)
6664
6665
# CE to independence_number(x) <= card_periphery(x) + floor(e^laplacian_energy(x))
6666
ce116 = Graph("J?Bzvrw}Fo?")
6667
ce116.name(new = "ce116")
6668
add_to_lists(ce116, graph_objects, counter_examples, all_graphs)
6669
6670
# CE to independence_number(x) <= residue(x)^2/density(x)
6671
ce117 = Graph("N???F~}~f{^_z_~_^o?")
6672
ce117.name(new = "ce117")
6673
add_to_lists(ce117, graph_objects, counter_examples, all_graphs)
6674
6675
# CE to independence_number(x) <= maximum(caro_wei(x), e^different_degrees(x))^2
6676
ce118 = Graph("O????B~~v}^w~o~o^wF}?")
6677
ce118.name(new = "ce118")
6678
add_to_lists(ce118, graph_objects, counter_examples, all_graphs)
6679
6680
# CE to independence_number(x) <= 10^sinh(average_vertex_temperature(x) + card_center(x))
6681
ce119 = Graph("\?@CoO_CDOC?G?_?O?CO?_?BO?C???O?Bw??k??D_??O???_???C???A???@????g????")
6682
ce119.name(new = "ce119")
6683
add_to_lists(ce119, graph_objects, counter_examples, all_graphs)
6684
6685
# CE to independence_number(x) <= maximum(girth(x), min_degree(x))^laplacian_energy(x)
6686
ce120 = Graph("^????????????????????X~~Umf|mezrBcjoezwF^{_Un}?|w{?kQ[?@Z}?FNl??Etw??Oz??F~Z_??")
6687
ce120.name(new = "ce120")
6688
add_to_lists(ce120, graph_objects, counter_examples, all_graphs)
6689
6690
# CE to independence_number(x) <= (2*girth(x))^card_periphery(x)
6691
ce121 = Graph("~?@HhCGGC@?G?_@?@??_?G?@??C??G??G??C??@???G???_??@???@????_???G???@????C????G????G????C????@?????G?????_????@?????@??????_?????G?????@??????C??????G??????G??????C??????@???????G???????_??????@???????@????????_???????G???????@????????C????????G????????G????????C????????@?????????G?????????_????????@?????????@??????????_?????????G?????????@??????????C??????????G??????????G??????????C??????????@???????????G???????????_??????????@???????????@")
6692
ce121.name(new = "ce121")
6693
add_to_lists(ce121, graph_objects, counter_examples, all_graphs)
6694
6695
# CE to independence_number(x) <= ceil(laplacian_energy(x)*max_degree(x))
6696
ce122 = Graph("~?@B??????????????????????????????????????????????????????????????????????????????????B\jeRlsk[rya~Sdr[HeojcW}xwcX_TgTVa?UhcYC?BTaaSo?}YoCRo@ov@_b?EysEu}?Do`uhm?@Ebfkm??~AHGh_?Awyl{w??TrUZDg??AuxBGO??FJ{LUo??O_IF]g??EVmFSY???`YcP_????l|v`}???Bg_[ge???ERCdO[???APuTc^????NQUWCO???B[QrBO????BMP^A_????eo{v`_????NgB\A_????O[Q@HO????@V_oTC?????FCDLPo????AECLPI?????DqZqCE??????")
6697
ce122.name(new = "ce122")
6698
add_to_lists(ce122, graph_objects, counter_examples, all_graphs)
6699
6700
# CE to independence_number(x) <= minimum(cvetkovic(x), 10^laplacian_energy(x))
6701
ce123 = Graph("`??????????????????????????????????????????^^}nx~~[~~x}^k~~F~v^o^~vm@^~~wF~v}_F~~^_B~~~W?")
6702
ce123.name(new = "ce123")
6703
add_to_lists(ce123, graph_objects, counter_examples, all_graphs)
6704
6705
# CE to independence_number(x) <= minimum(fractional_alpha(x), 10^laplacian_energy(x))
6706
ce124 = Graph("Z????????????????????~~~~~v~vf|~b~~o~~sF~~_^~}?~|k?z~w?^~}??")
6707
ce124.name(new = "ce124")
6708
add_to_lists(ce124, graph_objects, counter_examples, all_graphs)
6709
6710
# CE to independence_number(x) <= ceil(e^caro_wei(x))
6711
ce125 = Graph("L??F~z{~FwN_~?")
6712
ce125.name(new = "ce125")
6713
add_to_lists(ce125, graph_objects, counter_examples, all_graphs)
6714
6715
# CE to independence_number(x) <= ceil(e^inverse_degree(x))
6716
ce126 = Graph("Ov~~~}Lvs]^~~~~t~~}yJ")
6717
ce126.name(new = "ce126")
6718
add_to_lists(ce126, graph_objects, counter_examples, all_graphs)
6719
6720
# CE to independence_number(x) <= minimum(annihilation_number(x), 10^laplacian_energy(x))
6721
ce127 = Graph("[??????~~~^{~w~w^{F~?~WB~_F~?F~?B~_?~w?F~??^s??~w??~w??^{??F~???")
6722
ce127.name(new = "ce127")
6723
add_to_lists(ce127, graph_objects, counter_examples, all_graphs)
6724
6725
# CE to independence_number(x) <= (diameter(x) + 1)*residue(x)
6726
ce128 = Graph("M????B~~v}^w~o~o?")
6727
ce128.name(new = "ce128")
6728
add_to_lists(ce128, graph_objects, counter_examples, all_graphs)
6729
6730
# CE to independence_number(x) <= floor(10^sqrt(average_distance(x)))
6731
ce129 = Graph("X????????????????????????????F~~~~z~~~Z~~{n}|u~utn~")
6732
ce129.name(new = "ce129")
6733
add_to_lists(ce129, graph_objects, counter_examples, all_graphs)
6734
6735
# CE to independence_number(x) <= minimum(lovasz_theta(x), 10^laplacian_energy(x))
6736
ce130 = Graph("]?????????????????F|~nzz~~f~~F|~B~~_~~WF~~?^~{?nng?~~w?^^{?F~n??~~w?@~~_??")
6737
ce130.name(new = "ce130")
6738
add_to_lists(ce130, graph_objects, counter_examples, all_graphs)
6739
6740
# CE to independence_number(x) <= (residue(x) + 1)*girth(x)
6741
ce131 = Graph("W?????????^~~}~}^~F~o~}B~wF}oF~oB~w?~}?F~o?^~??")
6742
ce131.name(new = "ce131")
6743
add_to_lists(ce131, graph_objects, counter_examples, all_graphs)
6744
6745
# CE to independence_number(x) <= minimum(size(x), 10^laplacian_energy(x))
6746
ce132 = Graph("b????????????????????~~|~~|}~x~n~vp~{Nz{^e~svN}Xc~zFKZ~uyX[z[J\\ti@M~mT@]}X{SzQZQQ^posGNlZg?R}lp`LX{L?")
6747
ce132.name(new = "ce132")
6748
add_to_lists(ce132, graph_objects, counter_examples, all_graphs)
6749
6750
# CE to independence_number(x) >= -brinkmann_steffen(x) - girth(x) + matching_number(x)
6751
ce133 = Graph("~?@VNtyzkW^v~_?UMVuzfYa_~}|tWZ}v~~v}WgheAI[pTRjr\\Wv}y~VeypzF\\^zmq}C\\UFoD^XJFQwvU^AovUND~XnF]STE_EF~v^~~JF\\X{X\\VFoeMvmy~xzZv\\^{|n~v~~~^v}y~xz]Wihy_Lzmq]CxSawvUNFZr]zj~fl|vMy~wz^Z}q}MxvsZv\\^{|nnuMy~wr^\\vmy~xz^^z~\\^}|nny{qUMHRpZSSE_EC_A@VXJAShyGe|vV~NZz~i}vUNnZr~n]zj~fl|vr^u~p~~}~~z}v}N~~v~~nzNwzf^\\n~TuQodI]AHvuMy^wr^\\wkNz^w~|~^}~y}vUNNZr~n~v}y~|z^^|~~^zn~~n~~v~y}q]MxVtZn|y{qUMHRpZfx|WyhIbL|vap[~}~~~~~~~~~}|vV~NZz~fz~jzHWydNDmzvz~~~~~~~~~~~~c?AGgA]Kw_PC_?_???O?????V~v~~~^~~n~~}\\uRofIca@FHcd~z~N~~~~~~~}Zaw}?oscOOx?d}ptR|EZznDbxsj~z~~~~~~~~~}^cAAYgA^KwgPCc]~v^~~^~~n~~}znz\\w~|~^}~~~r|v}y~|z^^|~~~t^Vuyp|z]^|~~~dZz~\\^{|nn}~~~ynw")
6752
ce133.name(new = "ce133")
6753
add_to_lists(ce133, graph_objects, counter_examples, all_graphs)
6754
6755
# CE to independence_number(x) >= -max_common_neighbors(x) + min_degree(x) - 1
6756
ce134 = Graph('rQjxS{d~tvtrkz~pIJZ{gflAUdYPfybdzswqqS~i`~EfCmswgpu[zfAxLl\TtJEFlilHnZicmo}ZYJjAluT]d|scS\LgJ[|cs~}TBXxNQnJxSm]}oSMt{\kxUl|UhPZHlz`smizxCPTiNL[Mv|kbKUI}^r}oiAdMLE\^rga{v]z@U]Hb}wupjLh`Yg|Rn|`b[iLNp}Oudo~r_`oFEjTzvw')
6757
ce134.name(new = "ce134")
6758
add_to_lists(ce134, graph_objects, counter_examples, all_graphs)
6759
6760
# CE to independence_number(x) >= floor(tan(brooks(x)))
6761
ce135 = Graph('Nzx~VT}yzxNd^J^Jn~w')
6762
ce135.name(new = "ce135")
6763
add_to_lists(ce135, graph_objects, counter_examples, all_graphs)
6764
6765
# CE to independence_number(x) >= floor(lovasz_theta(x))*sin(gutman_energy(x))
6766
paley_17 = graphs.PaleyGraph(17)
6767
paley_17.name(new = "paley_17")
6768
add_to_lists(paley_17, graph_objects, counter_examples, all_graphs)
6769
6770
# CE to independence_number(x) >= ceil(lovasz_theta(x)) - diameter(x)
6771
# CE to independence_number(x) >= ceil(lovasz_theta(x)) - radius(x)
6772
paley_37 = graphs.PaleyGraph(37)
6773
paley_37.name(new = "paley_37")
6774
add_to_lists(paley_37, graph_objects, counter_examples, all_graphs)
6775
6776
# CE to independence_number(x) >= minimum(floor(lovasz_theta(x)), tan(randic(x)))
6777
paley_53e = graphs.PaleyGraph(53)
6778
paley_53e.add_edge(52,53)
6779
paley_53e.name(new = "paley_53e")
6780
add_to_lists(paley_53e, graph_objects, counter_examples, all_graphs)
6781
6782
paley_101 = graphs.PaleyGraph(101)
6783
paley_101.name(new = "paley_101")
6784
add_to_lists(paley_101, graph_objects, all_graphs)
6785
6786
# CE to independence_number(x) >= cos(alon_spencer(x))*floor(lovasz_theta(x))
6787
paley_149 = graphs.PaleyGraph(149)
6788
paley_149.name(new = "paley_149")
6789
add_to_lists(paley_149, graph_objects, counter_examples, all_graphs)
6790
6791
#a K5 with a pendant
6792
# CE to dirac => regular or planar
6793
k5pendant = Graph('E~}?')
6794
k5pendant.name(new="k5pendant")
6795
add_to_lists(k5pendant, graph_objects, counter_examples, all_graphs)
6796
6797
#alon_seymour graph:
6798
# CE to the rank-coloring conjecture,
6799
# 56-regular, vertex_trans, alpha=2, omega=22, chi=chi'=edge_connect=56
6800
alon_seymour=Graph([range(64), lambda x,y : operator.xor(x,y) not in (0,1,2,4,8,16,32,63)])
6801
alon_seymour.name(new="alon_seymour")
6802
add_to_lists(alon_seymour, problem_graphs, counter_examples, all_graphs)
6803
6804
edge_critical_5=graphs.CycleGraph(5)
6805
edge_critical_5.add_edge(0,3)
6806
edge_critical_5.add_edge(1,4)
6807
edge_critical_5.name(new="edge_critical_5")
6808
add_to_lists(edge_critical_5, graph_objects, chromatic_index_critical, all_graphs)
6809
6810
# CE to independence_number(x) >= min(e - n + 1, diameter(x))
6811
heather = graphs.CompleteGraph(4)
6812
heather.add_vertex()
6813
heather.add_vertex()
6814
heather.add_edge(0,4)
6815
heather.add_edge(5,4)
6816
heather.name(new="heather")
6817
add_to_lists(heather, graph_objects, counter_examples, all_graphs)
6818
6819
#residue = alpha = 3
6820
# CE to residue = alpha => is_ore
6821
ryan3=graphs.CycleGraph(15)
6822
for i in range(15):
6823
for j in [1,2,3]:
6824
ryan3.add_edge(i,(i+j)%15)
6825
ryan3.add_edge(i,(i-j)%15)
6826
ryan3.name(new="ryan3")
6827
add_to_lists(ryan3, graph_objects, counter_examples, all_graphs)
6828
6829
#sylvester graph: 3-reg, 3 bridges, no perfect matching (why Petersen theorem requires no more than 2 bridges)
6830
sylvester = Graph('Olw?GCD@o??@?@?A_@o`A')
6831
sylvester.name(new="sylvester")
6832
add_to_lists(sylvester, graph_objects, all_graphs)
6833
6834
star_1_1_3 = graphs.PathGraph(4)
6835
star_1_1_3.add_vertex()
6836
star_1_1_3.add_edge(1,4)
6837
star_1_1_3.name(new="fork")
6838
add_to_lists(star_1_1_3, graph_objects, all_graphs)
6839
6840
# one of the 2 order 11 chromatic edge-critical graphs discovered by brinkmann and steffen
6841
edge_critical_11_1 = graphs.CycleGraph(11)
6842
edge_critical_11_1.add_edge(0,2)
6843
edge_critical_11_1.add_edge(1,6)
6844
edge_critical_11_1.add_edge(3,8)
6845
edge_critical_11_1.add_edge(5,9)
6846
edge_critical_11_1.name(new="edge_critical_11_1")
6847
add_to_lists(edge_critical_11_1, graph_objects, chromatic_index_critical, all_graphs)
6848
6849
#one of the 2 order 11 chromatic edge-critical graphs discovered by brinkmann and steffen
6850
edge_critical_11_2 = graphs.CycleGraph(11)
6851
edge_critical_11_2.add_edge(0,2)
6852
edge_critical_11_2.add_edge(3,7)
6853
edge_critical_11_2.add_edge(6,10)
6854
edge_critical_11_2.add_edge(4,9)
6855
edge_critical_11_2.name(new="edge_critical_11_2")
6856
add_to_lists(edge_critical_11_2, graph_objects, chromatic_index_critical, all_graphs)
6857
6858
# chromatic_index_critical but not overfull
6859
pete_minus=graphs.PetersenGraph()
6860
pete_minus.delete_vertex(9)
6861
pete_minus.name(new="pete_minus")
6862
add_to_lists(pete_minus, graph_objects, chromatic_index_critical, all_graphs)
6863
6864
"""
6865
The Haemers graph was considered by Haemers who showed that alpha(G)=theta(G)<vartheta(G).
6866
The graph is a 108-regular graph on 220 vertices. The vertices correspond to the 3-element
6867
subsets of {1,...,12} and two such vertices are adjacent whenever the subsets
6868
intersect in exactly one element.
6869
6870
sage: haemers
6871
haemers: Graph on 220 vertices
6872
sage: haemers.is_regular()
6873
True
6874
sage: max(haemers.degree())
6875
108
6876
"""
6877
haemers = Graph([Subsets(12,3), lambda s1,s2: len(s1.intersection(s2))==1])
6878
haemers.relabel()
6879
haemers.name(new="haemers")
6880
add_to_lists(haemers, problem_graphs, all_graphs)
6881
6882
"""
6883
The Pepper residue graph was described by Ryan Pepper in personal communication.
6884
It is a graph which demonstrates that the residue is not monotone. The graph is
6885
constructed by taking the complete graph on 3 vertices and attaching a pendant
6886
vertex to each of its vertices, then taking two copies of this graph, adding a
6887
vertex and connecting it to all the pendant vertices. This vertex has degree
6888
sequence [6, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2] which gives residue equal to 4.
6889
By removing the central vertex with degree 6, you get a graph with degree
6890
sequence [3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1] which has residue equal to 5.
6891
6892
sage: pepper_residue_graph
6893
pepper_residue_graph: Graph on 13 vertices
6894
sage: sorted(pepper_residue_graph.degree(), reverse=True)
6895
[6, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2]
6896
sage: residue(pepper_residue_graph)
6897
4
6898
sage: residue(pepper_residue_graph.subgraph(vertex_property=lambda v:pepper_residue_graph.degree(v)<6))
6899
5
6900
"""
6901
pepper_residue_graph = graphs.CompleteGraph(3)
6902
pepper_residue_graph.add_edges([(i,i+3) for i in range(3)])
6903
pepper_residue_graph = pepper_residue_graph.disjoint_union(pepper_residue_graph)
6904
pepper_residue_graph.add_edges([(0,v) for v in pepper_residue_graph.vertices() if pepper_residue_graph.degree(v)==1])
6905
pepper_residue_graph.relabel()
6906
pepper_residue_graph.name(new="pepper_residue_graph")
6907
add_to_lists(pepper_residue_graph, graph_objects, all_graphs)
6908
6909
"""
6910
The Barrus graph was suggested by Mike Barrus in "Havel-Hakimi residues of Unigraphs" (2012) as an example of a graph whose residue (2) is
6911
less than the independence number of any realization of the degree sequence. The degree sequence is [4^8,2].
6912
The realization is the one given by reversing the Havel-Hakimi process.
6913
6914
sage: barrus_graph
6915
barrus_graph: Graph on 9 vertices
6916
sage: residue(barrus_graph)
6917
2
6918
sage: independence_number(barrus_graph)
6919
3
6920
"""
6921
barrus_graph = Graph('HxNEG{W')
6922
barrus_graph.name(new = "barrus_graph")
6923
add_to_lists(barrus_graph, graph_objects, all_graphs)
6924
6925
# CE to (is_split)->((is_eulerian)->(is_regular))
6926
# split graph from k4 and e2 that is eulerian but not regular
6927
# isomorphic to c6 with a k4 subgraph
6928
# eulerain, diameter = 3, radius=2, hamiltonian
6929
6930
k4e2split = graphs.CompleteGraph(4)
6931
k4e2split.add_vertices([4,5])
6932
k4e2split.add_edge(4,0)
6933
k4e2split.add_edge(4,1)
6934
k4e2split.add_edge(5,2)
6935
k4e2split.add_edge(5,3)
6936
k4e2split.name(new = "k4e2split")
6937
add_to_lists(k4e2split, graph_objects, counter_examples, all_graphs)
6938
6939
# CE to (has_residue_equals_alpha)->((is_eulerian)->(alpha_leq_order_over_two))
6940
triangle_star = Graph("H}qdB@_")
6941
triangle_star.name(new = "triangle_star")
6942
add_to_lists(triangle_star, graph_objects, counter_examples, all_graphs)
6943
6944
#flower with n petals
6945
def flower(n):
6946
g = graphs.StarGraph(2*n)
6947
for x in range(n):
6948
v = 2*x+1
6949
g.add_edge(v,v+1)
6950
return g
6951
6952
flower_with_3_petals = flower(3)
6953
flower_with_3_petals.name(new = "flower_with_3_petals")
6954
add_to_lists(flower_with_3_petals, graph_objects, all_graphs)
6955
6956
flower_with_4_petals = flower(4)
6957
flower_with_4_petals.name(new = "flower_with_4_petals")
6958
add_to_lists(flower_with_4_petals, graph_objects, all_graphs)
6959
6960
# Gallai Tree graph
6961
gallai_tree = Graph("`hCKGC@?G@?K?@?@_?w?@??C??G??G??c??o???G??@_??F???N????_???G???B????C????W????G????G????C")
6962
gallai_tree.name(new = "gallai_tree")
6963
add_to_lists(gallai_tree, graph_objects, all_graphs)
6964
6965
# Trigonal Antiprism w/ capped top face
6966
trig_antiprism_capped = Graph("Iw?EthkF?")
6967
trig_antiprism_capped.name(new = "trig_antiprism_capped")
6968
add_to_lists(trig_antiprism_capped, graph_objects, all_graphs)
6969
6970
"""
6971
From Willis's thesis, page 4
6972
Alpha = Fractional Alpha = 4
6973
6974
sage: independence_number(willis_page4)
6975
4
6976
sage: fractional_alpha(willis_page4)
6977
4.0
6978
"""
6979
willis_page4 = Graph("GlCKIS")
6980
willis_page4.name(new = "willis_page4")
6981
add_to_lists(willis_page4, graph_objects, all_graphs)
6982
6983
"""
6984
From Willis's thesis, page 7
6985
6986
sage: willis_page7.radius()
6987
2
6988
sage: willis_page7.average_distance()
6989
22/15
6990
"""
6991
willis_page7 = Graph("ELRW")
6992
willis_page7.name(new = "willis_page7")
6993
add_to_lists(willis_page7, graph_objects, all_graphs)
6994
6995
"""
6996
From Willis's thesis, page 13, Fig. 2.7
6997
6998
sage: independence_number(willis_page13_fig27)
6999
3
7000
sage: willis_page13_fig27.order()
7001
7
7002
sage: willis_page13_fig27.size()
7003
15
7004
"""
7005
willis_page13_fig27 = Graph("Fs\zw")
7006
willis_page13_fig27.name(new = "willis_page13_fig27")
7007
add_to_lists(willis_page13_fig27, graph_objects, all_graphs)
7008
7009
"""
7010
From Willis's thesis, page 10, Figure 2.3
7011
Graph for which the Cvetkovic bound is the best upper bound present in the thesis
7012
7013
sage: independence_number(willis_page10_fig23)
7014
3
7015
sage: willis_page10_fig23.order()
7016
8
7017
sage: willis_page10_fig23.size()
7018
15
7019
sage: max_degree(willis_page10_fig23)
7020
6
7021
sage: min_degree(willis_page10_fig23)
7022
2
7023
"""
7024
willis_page10_fig23 = Graph("G|eKHw")
7025
willis_page10_fig23.name(new = "willis_page10_fig23")
7026
add_to_lists(willis_page10_fig23, graph_objects, all_graphs)
7027
7028
"""
7029
From Willis's thesis, page 10, Figure 2.4
7030
Graph for which the Cvetkovic bound is the best upper bound present in the thesis
7031
7032
sage: independence_number(willis_page10_fig24)
7033
9
7034
sage: willis_page10_fig24.order()
7035
24
7036
sage: willis_page10_fig24.size()
7037
36
7038
sage: max_degree(willis_page10_fig24)
7039
3
7040
sage: min_degree(willis_page10_fig24)
7041
3
7042
"""
7043
willis_page10_fig24 = Graph("WvOGWK@?G@_B???@_?O?F?????G??W?@K_?????G??@_?@B")
7044
willis_page10_fig24.name(new = "willis_page10_fig24")
7045
add_to_lists(willis_page10_fig24, graph_objects, all_graphs)
7046
7047
"""
7048
From Willis's thesis, page 13, Figure 2.6
7049
Graph for which the fractional independence bound is the best upper bound present in the thesis
7050
7051
sage: independence_number(willis_page13_fig26)
7052
3
7053
sage: willis_page13_fig26.order()
7054
7
7055
sage: willis_page13_fig26.size()
7056
12
7057
sage: max_degree(willis_page13_fig26)
7058
4
7059
sage: min_degree(willis_page13_fig26)
7060
3
7061
"""
7062
willis_page13_fig26 = Graph("FstpW")
7063
willis_page13_fig26.name(new = "willis_page13_fig26")
7064
add_to_lists(willis_page13_fig26, graph_objects, all_graphs)
7065
7066
"""
7067
From Willis's thesis, page 21, Figure 3.1
7068
Graph for which n/chi is the best lower bound present in the thesis
7069
7070
sage: independence_number(willis_page21)
7071
4
7072
sage: willis_page21.order()
7073
12
7074
sage: willis_page21.size()
7075
20
7076
sage: max_degree(willis_page21)
7077
4
7078
sage: willis_page21.chromatic_number()
7079
3
7080
"""
7081
willis_page21 = Graph("KoD?Xb?@hBBB")
7082
willis_page21.name(new = "willis_page21")
7083
add_to_lists(willis_page21, graph_objects, all_graphs)
7084
7085
"""
7086
From Willis's thesis, page 25, Figure 3.2
7087
Graph for which residue is the best lower bound present in the thesis
7088
7089
sage: independence_number(willis_page25_fig32)
7090
3
7091
sage: willis_page25_fig32.order()
7092
8
7093
sage: willis_page25_fig32.size()
7094
15
7095
sage: max_degree(willis_page25_fig32)
7096
6
7097
sage: willis_page25_fig32.chromatic_number()
7098
4
7099
"""
7100
willis_page25_fig32 = Graph("G@N@~w")
7101
willis_page25_fig32.name(new = "willis_page25_fig32")
7102
add_to_lists(willis_page25_fig32, graph_objects, all_graphs)
7103
7104
"""
7105
From Willis's thesis, page 25, Figure 3.3
7106
Graph for which residue is the best lower bound present in the thesis
7107
7108
sage: independence_number(willis_page25_fig33)
7109
4
7110
sage: willis_page25_fig33.order()
7111
14
7112
sage: willis_page25_fig33.size()
7113
28
7114
sage: max_degree(willis_page25_fig33)
7115
4
7116
sage: willis_page25_fig33.chromatic_number()
7117
4
7118
"""
7119
willis_page25_fig33 = Graph("Mts?GKE@QDCIQIKD?")
7120
willis_page25_fig33.name(new = "willis_page25_fig33")
7121
add_to_lists(willis_page25_fig33, graph_objects, all_graphs)
7122
7123
# The Lemke Graph
7124
lemke = Graph("G_?ztw")
7125
lemke.name(new = "Lemke")
7126
add_to_lists(lemke, graph_objects, all_graphs)
7127
7128
"""
7129
From Willis's thesis, page 29, Figure 3.6
7130
Graph for which the Harant Bound is the best lower bound present in the thesis
7131
7132
sage: independence_number(willis_page29)
7133
13
7134
sage: willis_page29.order()
7135
28
7136
sage: willis_page29.size()
7137
42
7138
sage: max_degree(willis_page29)
7139
3
7140
sage: willis_page29.chromatic_number()
7141
3
7142
"""
7143
willis_page29 = mobius_ladder(14)
7144
willis_page29.name(new = "willis_page29")
7145
add_to_lists(willis_page29, graph_objects, all_graphs)
7146
7147
"""
7148
From Willis's thesis, page 35, Figure 5.1
7149
A graph where none of the upper bounds in the thesis give the exact value for alpha
7150
7151
sage: independence_number(willis_page35_fig51)
7152
2
7153
sage: willis_page35_fig51.order()
7154
10
7155
"""
7156
willis_page35_fig51 = Graph("I~rH`cNBw")
7157
willis_page35_fig51.name(new = "willis_page35_fig51")
7158
add_to_lists(willis_page35_fig51, graph_objects, all_graphs)
7159
7160
"""
7161
From Willis's thesis, page 35, Figure 5.2
7162
A graph where none of the upper bounds in the thesis give the exact value for alpha
7163
7164
sage: independence_number(willis_page35_fig52)
7165
2
7166
sage: willis_page35_fig52.order()
7167
10
7168
"""
7169
willis_page35_fig52 = Graph("I~zLa[vFw")
7170
willis_page35_fig52.name(new = "willis_page35_fig52")
7171
add_to_lists(willis_page35_fig52, graph_objects, all_graphs)
7172
7173
"""
7174
From Willis's thesis, page 36, Figure 5.3
7175
A graph where none of the upper bounds in the thesis give the exact value for alpha
7176
7177
sage: independence_number(willis_page36_fig53)
7178
4
7179
sage: willis_page36_fig53.order()
7180
11
7181
"""
7182
willis_page36_fig53 = Graph("JscOXHbWqw?")
7183
willis_page36_fig53.name(new = "willis_page36_fig53")
7184
add_to_lists(willis_page36_fig53, graph_objects, all_graphs)
7185
7186
"""
7187
From Willis's thesis, page 36, Figure 5.4
7188
A graph where none of the upper bounds in the thesis give the exact value for alpha
7189
7190
sage: independence_number(willis_page36_fig54)
7191
2
7192
sage: willis_page36_fig54.order()
7193
9
7194
sage: willis_page36_fig54.size()
7195
24
7196
"""
7197
willis_page36_fig54 = Graph("H~`HW~~")
7198
willis_page36_fig54.name(new = "willis_page36_fig54")
7199
add_to_lists(willis_page36_fig54, graph_objects, all_graphs)
7200
7201
"""
7202
From Willis's thesis, page 36, Figure 5.5
7203
A graph where none of the lower bounds in the thesis give the exact value for alpha
7204
7205
sage: independence_number(willis_page36_fig55)
7206
3
7207
sage: willis_page36_fig55.order()
7208
7
7209
sage: willis_page36_fig55.size()
7210
13
7211
"""
7212
willis_page36_fig55 = Graph("F@^vo")
7213
willis_page36_fig55.name(new = "willis_page36_fig55")
7214
add_to_lists(willis_page36_fig55, graph_objects, all_graphs)
7215
7216
"""
7217
From Willis's thesis, page 37, Figure 5.6
7218
A graph where none of the lower bounds in the thesis give the exact value for alpha
7219
7220
sage: independence_number(willis_page37_fig56)
7221
3
7222
sage: willis_page37_fig56.order()
7223
7
7224
sage: willis_page37_fig56.size()
7225
15
7226
"""
7227
willis_page37_fig56 = Graph("Fimzw")
7228
willis_page37_fig56.name(new = "willis_page37_fig56")
7229
add_to_lists(willis_page37_fig56, graph_objects, all_graphs)
7230
7231
"""
7232
From Willis's thesis, page 37, Figure 5.8
7233
A graph where none of the lower bounds in the thesis give the exact value for alpha
7234
7235
sage: independence_number(willis_page37_fig58)
7236
4
7237
sage: willis_page37_fig58.order()
7238
9
7239
sage: willis_page37_fig58.size()
7240
16
7241
sage: min_degree(willis_page37_fig58)
7242
2
7243
sage: max_degree(willis_page37_fig58)
7244
6
7245
"""
7246
willis_page37_fig58 = Graph("H?iYbC~")
7247
willis_page37_fig58.name(new = "willis_page37_fig58")
7248
add_to_lists(willis_page37_fig58, graph_objects, all_graphs)
7249
7250
"""
7251
From Willis's thesis, page 38, Figure 5.9
7252
A graph where none of the lower bounds in the thesis give the exact value for alpha
7253
7254
sage: independence_number(willis_page38)
7255
3
7256
sage: willis_page38.order()
7257
7
7258
sage: willis_page38.size()
7259
13
7260
"""
7261
willis_page38 = Graph("FSpzw")
7262
willis_page38.name(new = "willis_page38")
7263
add_to_lists(willis_page38, graph_objects, all_graphs)
7264
7265
"""
7266
From Willis's thesis, page 39, Figure 5.10
7267
A graph where none of the upper or lower bounds in the thesis give the exact value for alpha
7268
7269
sage: independence_number(willis_page39_fig510)
7270
5
7271
sage: willis_page39_fig510.order()
7272
12
7273
sage: willis_page39_fig510.size()
7274
18
7275
"""
7276
willis_page39_fig510 = Graph("Kt?GOKEOGal?")
7277
willis_page39_fig510.name(new = "willis_page39_fig510")
7278
add_to_lists(willis_page39_fig510, graph_objects, all_graphs)
7279
7280
"""
7281
From Willis's thesis, page 40, Figure 5.12
7282
A graph where none of the upper or lower bounds in the thesis give the exact value for alpha
7283
7284
sage: independence_number(willis_page40_fig512)
7285
6
7286
sage: willis_page40_fig512.order()
7287
14
7288
sage: willis_page40_fig512.size()
7289
21
7290
"""
7291
willis_page40_fig512 = Graph("Ms???\?OGdAQJ?J??")
7292
willis_page40_fig512.name(new = "willis_page40_fig512")
7293
add_to_lists(willis_page40_fig512, graph_objects, all_graphs)
7294
7295
"""
7296
From Willis's thesis, page 40, Figure 5.13
7297
A graph where none of the upper or lower bounds in the thesis give the exact value for alpha
7298
"""
7299
willis_page40_fig513 = Graph("UhCGKE??Q?_D?K??__g?@G?C??OC???ACG?DC??_")
7300
willis_page40_fig513.name(new = "willis_page40_fig513")
7301
add_to_lists(willis_page40_fig513, graph_objects, all_graphs)
7302
7303
"""
7304
From Willis's thesis, page 41, Figure 5.14
7305
A graph where none of the upper or lower bounds in the thesis give the exact value for alpha
7306
7307
sage: independence_number(willis_page41_fig514)
7308
5
7309
sage: willis_page41_fig514.order()
7310
12
7311
sage: willis_page41_fig514.size()
7312
18
7313
"""
7314
willis_page41_fig514 = Graph("Kt?GGGBQGeL?")
7315
willis_page41_fig514.name(new = "willis_page41_fig514")
7316
add_to_lists(willis_page41_fig514, graph_objects, all_graphs)
7317
7318
"""
7319
From Willis's thesis, page 41, Figure 5.15
7320
A graph where none of the upper or lower bounds in the thesis give the exact value for alpha
7321
7322
sage: independence_number(willis_page41_fig515)
7323
4
7324
sage: willis_page41_fig515.order()
7325
11
7326
sage: willis_page41_fig515.size()
7327
22
7328
"""
7329
willis_page41_fig515 = Graph("JskIIDBLPh?")
7330
willis_page41_fig515.name(new = "willis_page41_fig515")
7331
add_to_lists(willis_page41_fig515, graph_objects, all_graphs)
7332
7333
"""
7334
From Elphick-Wocjan page 8
7335
"""
7336
elphick_wocjan_page8 = Graph("F?Azw")
7337
elphick_wocjan_page8.name(new = "Elphick-Wocjan p.8")
7338
add_to_lists(elphick_wocjan_page8, graph_objects, all_graphs)
7339
7340
"""
7341
From Elphick-Wocjan page 9
7342
"""
7343
elphick_wocjan_page9 = Graph("FqhXw")
7344
elphick_wocjan_page9.name(new = "Elphick-Wocjan p.9")
7345
add_to_lists(elphick_wocjan_page9, graph_objects, all_graphs)
7346
7347
"""
7348
An odd wheel with 8 vertices
7349
p.175
7350
Rebennack, Steffen, Gerhard Reinelt, and Panos M. Pardalos. "A tutorial on branch and cut algorithms for the maximum stable set problem." International Transactions in Operational Research 19.1-2 (2012): 161-199.
7351
7352
sage: odd_wheel_8.order()
7353
8
7354
sage: odd_wheel_8.size()
7355
14
7356
"""
7357
odd_wheel_8 = Graph("G|eKMC")
7358
odd_wheel_8.name(new = "odd_wheel_8")
7359
add_to_lists(odd_wheel_8, graph_objects, all_graphs)
7360
7361
"""
7362
A facet-inducing graph
7363
p.176
7364
Rebennack, Steffen, Gerhard Reinelt, and Panos M. Pardalos. "A tutorial on branch and cut algorithms for the maximum stable set problem." International Transactions in Operational Research 19.1-2 (2012): 161-199.
7365
7366
sage: facet_inducing.order()
7367
8
7368
sage: facet_inducing.size()
7369
11
7370
"""
7371
facet_inducing = Graph("G@hicc")
7372
facet_inducing.name(new = "facet_inducing")
7373
add_to_lists(facet_inducing, graph_objects, all_graphs)
7374
7375
"""
7376
Double Fork
7377
p.185
7378
Rebennack, Steffen, Gerhard Reinelt, and Panos M. Pardalos. "A tutorial on branch and cut algorithms for the maximum stable set problem." International Transactions in Operational Research 19.1-2 (2012): 161-199.
7379
7380
sage: double_fork.order()
7381
6
7382
sage: double_fork.size()
7383
5
7384
"""
7385
double_fork = Graph("E?dg")
7386
double_fork.name(new = "double_fork")
7387
add_to_lists(double_fork, graph_objects, all_graphs)
7388
7389
double_fork_sub = Graph("G??MPg")
7390
double_fork_sub.name(new = "double_fork_sub")
7391
add_to_lists(double_fork_sub, graph_objects, all_graphs)
7392
7393
"""
7394
Golomb Graph
7395
Appears in THE FRACTIONAL CHROMATIC NUMBER OF THE PLANE by Cranston and Rabern
7396
"""
7397
golomb = Graph("I?C]dPcww")
7398
golomb.name(new = "Golomb Graph")
7399
add_to_lists(golomb, graph_objects, all_graphs)
7400
7401
"""
7402
Fig. 1, p. 2 of
7403
Steinberg’s Conjecture is false by
7404
Vincent Cohen-Addad, Michael Hebdige, Daniel Král,
7405
Zhentao Li, Esteban Salgado
7406
"""
7407
steinberg_ce_g1 = Graph("N?CWGOOOH@OO_POdCHO")
7408
steinberg_ce_g1.name(new = "steinberg_ce_g1")
7409
add_to_lists(steinberg_ce_g1, graph_objects, counter_examples, all_graphs)
7410
7411
"""
7412
Fig. 2, p. 3 of
7413
Steinberg’s Conjecture is false by
7414
Vincent Cohen-Addad, Michael Hebdige, Daniel Král,
7415
Zhentao Li, Esteban Salgado
7416
"""
7417
steinberg_ce_g2 = Graph("ihCGGC@?G?o@o???_?H?@??K??Go?K?G??B???@gW?????G???P???K???@????P???@????B????BG????_?G??G?????G????A?O???H?????C????`?_???CO??????P?K????o@_????o")
7418
steinberg_ce_g2.name(new = "steinberg_ce_g2")
7419
add_to_lists(steinberg_ce_g2, graph_objects, counter_examples, all_graphs)
7420
7421
"""
7422
Fig. 3, p. 4 of
7423
Steinberg’s Conjecture is false by
7424
Vincent Cohen-Addad, Michael Hebdige, Daniel Král,
7425
Zhentao Li, Esteban Salgado
7426
"""
7427
steinberg_ce_g3 = Graph("mhE?GC@?G?`@AG?G_?G?@_?E?????GO?EG????O????`??@_??G????_???GO??@?????????G????GO???CG???GA??????????_????@????AHA???????????O????O???????O??????_??????_???C??O???A??C????_??_")
7428
steinberg_ce_g3.name(new = "steinberg_ce_g3")
7429
add_to_lists(steinberg_ce_g3, graph_objects, counter_examples, all_graphs)
7430
7431
"""
7432
4-pan from p. 1691 of
7433
Graphs with the Strong Havel–Hakimi Property by Michael D. Barrus and Grant Molnar
7434
"""
7435
four_pan = Graph("DBw")
7436
four_pan.name(new = "4-pan")
7437
add_to_lists(four_pan, graph_objects, all_graphs)
7438
7439
"""
7440
kite from p. 1691 of
7441
Graphs with the Strong Havel–Hakimi Property by Michael D. Barrus and Grant Molnar
7442
NOTE: Called kite in the paper, but will be called kite_with_tail here because we already have a kite
7443
"""
7444
kite_with_tail = Graph("DJk")
7445
kite_with_tail.name(new = "kite with tail")
7446
add_to_lists(kite_with_tail, graph_objects, all_graphs)
7447
7448
"""
7449
Chartrand Fig 1.1
7450
7451
sage: chartrand_11.order()
7452
8
7453
sage: chartrand_11.size()
7454
15
7455
"""
7456
chartrand_11 = Graph("G`RHx{")
7457
chartrand_11.name(new = "chartrand fig 1.1")
7458
add_to_lists(chartrand_11, graph_objects, all_graphs)
7459
7460
"""
7461
Chartrand Fig 1.2
7462
7463
sage: chartrand_12.order()
7464
8
7465
sage: chartrand_12.size()
7466
9
7467
"""
7468
chartrand_12 = Graph("G??|Qo")
7469
chartrand_12.name(new = "chartrand fig 1.2")
7470
add_to_lists(chartrand_12, graph_objects, all_graphs)
7471
7472
"""
7473
Chartrand Fig 1.3
7474
7475
sage: chartrand_13.order()
7476
8
7477
sage: chartrand_13.size()
7478
10
7479
"""
7480
chartrand_13 = Graph("G`o_g[")
7481
chartrand_13.name(new = "chartrand fig 1.3")
7482
add_to_lists(chartrand_13, graph_objects, all_graphs)
7483
7484
"""
7485
Chartrand Fig 1.8 - G
7486
7487
sage: chartrand_18_g.order()
7488
7
7489
sage: chartrand_18_g.size()
7490
8
7491
"""
7492
chartrand_18_g = Graph("Fo@Xo")
7493
chartrand_18_g.name(new = "chartrand fig 1.8 - G")
7494
add_to_lists(chartrand_18_g, graph_objects, all_graphs)
7495
7496
"""
7497
Chartrand Fig 1.8 - F1
7498
7499
sage: chartrand_18_f1.order()
7500
7
7501
sage: chartrand_18_f1.size()
7502
10
7503
"""
7504
chartrand_18_f1 = Graph("F@J]o")
7505
chartrand_18_f1.name(new = "chartrand fig 1.8 - F1")
7506
add_to_lists(chartrand_18_f1, graph_objects, all_graphs)
7507
7508
"""
7509
Chartrand Fig 1.8 - F2
7510
7511
sage: chartrand_18_f2.order()
7512
7
7513
sage: chartrand_18_f2.size()
7514
10
7515
"""
7516
chartrand_18_f2 = Graph("F?NVo")
7517
chartrand_18_f2.name(new = "chartrand fig 1.8 - F2")
7518
add_to_lists(chartrand_18_f2, graph_objects, all_graphs)
7519
7520
"""
7521
Chartrand Fig 1.10 - H1
7522
CE to independence_number(x) <= maximum(girth(x), card_center(x) + card_periphery(x))
7523
7524
sage: chartrand_110_h1.order()
7525
7
7526
sage: chartrand_110_h1.size()
7527
7
7528
"""
7529
chartrand_110_h1 = Graph("F@@Kw")
7530
chartrand_110_h1.name(new = "chartrand fig 1.10 - H1")
7531
add_to_lists(chartrand_110_h1, graph_objects, counter_examples, all_graphs)
7532
7533
"""
7534
Chartrand Fig 1.10 - H2
7535
7536
sage: chartrand_110_h2.order()
7537
7
7538
sage: chartrand_110_h2.size()
7539
7
7540
"""
7541
chartrand_110_h2 = Graph("F?C^G")
7542
chartrand_110_h2.name(new = "chartrand fig 1.10 - H2")
7543
add_to_lists(chartrand_110_h2, graph_objects, all_graphs)
7544
7545
"""
7546
Chartrand Fig 1.10 - H3
7547
7548
sage: chartrand_110_h3.order()
7549
9
7550
sage: chartrand_110_h3.size()
7551
8
7552
"""
7553
chartrand_110_h3 = Graph("H???]Os")
7554
chartrand_110_h3.name(new = "chartrand fig 1.10 - H3")
7555
add_to_lists(chartrand_110_h3, graph_objects, all_graphs)
7556
7557
"""
7558
Chartrand Fig 1.10 - H4
7559
7560
sage: chartrand_110_h4.order()
7561
9
7562
sage: chartrand_110_h4.size()
7563
8
7564
"""
7565
chartrand_110_h4 = Graph("H?C?JEK")
7566
chartrand_110_h4.name(new = "chartrand fig 1.10 - H4")
7567
add_to_lists(chartrand_110_h4, graph_objects, all_graphs)
7568
7569
# From their Mathing Theory book
7570
lovasz_plummer = Graph('iOQBC__???G_?OCG@??C???_C?G?@_?__??????_??@???E?C?C?A?A??CC????O???@??G??_????o?????????A?????O????B?????A????CO?????C?????@??????C??????W?????CO')
7571
lovasz_plummer.name(new = "lovasz_plummer graph")
7572
add_to_lists(lovasz_plummer, graph_objects, all_graphs)
7573
7574
# Jorgenson graphs from: Jørgensen, Leif K. "Diameters of cubic graphs." Discrete applied mathematics 37 (1992): 347-351.
7575
jorgenson_1 = Graph('Ss??GOG?OC?_?I?DGCa?oDO?i??_C@?AC')
7576
jorgenson_1.name(new = "jorgenson_1")
7577
add_to_lists(jorgenson_1, graph_objects, all_graphs)
7578
7579
jorgenson_2 = Graph('GtP@Ww')
7580
jorgenson_2.name(new = "jorgenson_2")
7581
add_to_lists(jorgenson_2, graph_objects, all_graphs)
7582
7583
# CGT graphs come from
7584
# Examples and Counter Examples in Graph Theory by Michael Capobianco and John C. Molluzzo
7585
7586
cgt_5 = Graph("J??GkPPgbG?")
7587
cgt_5.name(new = "CGT Page 5")
7588
add_to_lists(cgt_5, graph_objects, all_graphs)
7589
7590
cgt_10_left = Graph("E`~o")
7591
cgt_10_left.name(new = "CGT Page 10, left")
7592
add_to_lists(cgt_10_left, graph_objects, all_graphs)
7593
7594
cgt_10_right = Graph("EFxw")
7595
cgt_10_right.name(new = "CGT Page 10, right")
7596
add_to_lists(cgt_10_right, graph_objects, all_graphs)
7597
7598
cgt_11_1221 = Graph("H`_YPN~")
7599
cgt_11_1221.name(new = "CGT Page 11, Figure 1.22.1")
7600
add_to_lists(cgt_11_1221, graph_objects, all_graphs)
7601
7602
cgt_13_bottom = Graph("Iv?GOKFY?")
7603
cgt_13_bottom.name(new = "CGT Page 13, Bottom")
7604
add_to_lists(cgt_13_bottom, graph_objects, all_graphs)
7605
7606
cgt_15_left = Graph("FD^vW")
7607
cgt_15_left.name(new = "CGT Page 15, left")
7608
add_to_lists(cgt_15_left, graph_objects, all_graphs)
7609
7610
# Sabidussi Graphs from Sabidussi, Gert. "The centrality index of a graph." Psychometrika 31.4 (1966): 581-603.
7611
7612
sabidussi_2 = Graph('Ls_aGcKSpiDCQH')
7613
sabidussi_2.name(new = "sabidussi_2")
7614
add_to_lists(sabidussi_2, graph_objects, all_graphs)
7615
7616
sabidussi_3 = Graph('H`?NeW{')
7617
sabidussi_3.name(new = "sabidussi_3")
7618
add_to_lists(sabidussi_3, graph_objects, all_graphs)
7619
7620
sabidussi_4 = Graph('L?O_?C@?G_oOF?')
7621
sabidussi_4.name(new = "sabidussi_4")
7622
add_to_lists(sabidussi_4, graph_objects, all_graphs)
7623
7624
sabidussi_5 = Graph('L??GH?GOC@AWKC')
7625
sabidussi_5.name(new = "sabidussi_5")
7626
add_to_lists(sabidussi_5, graph_objects, all_graphs)
7627
7628
sabidussi_6 = Graph('L??G?COGQCOCN?')
7629
sabidussi_6.name(new = "sabidussi_6")
7630
add_to_lists(sabidussi_6, graph_objects, all_graphs)
7631
7632
barnette = Graph("Ss?GOCDA?@_I@??_C?q?QC?_O@@??I??S")
7633
barnette.name(new = "Barnette-Bosak-Lederberg Graph")
7634
add_to_lists(barnette, graph_objects, all_graphs)
7635
7636
# Maximally Irregular Graphs
7637
mir_6 = MIR(6)
7638
mir_6.name(new = "max_irregular_6")
7639
add_to_lists(mir_6, graph_objects, all_graphs)
7640
7641
mir_7 = MIR(7)
7642
mir_7.name(new = "max_irregular_7")
7643
add_to_lists(mir_7, graph_objects, all_graphs)
7644
7645
# Radius Critical Ciliates
7646
c4_1 = Ciliate(2, 3)
7647
c4_1.name(new = "Ciliate 4, 1")
7648
add_to_lists(c4_1, graph_objects, all_graphs)
7649
7650
c4_2 = Ciliate(2, 4)
7651
c4_2.name(new = "Ciliate 4, 2")
7652
add_to_lists(c4_2, graph_objects, all_graphs)
7653
7654
c6_1 = Ciliate(3, 4)
7655
c6_1.name(new = "Ciliate 6, 1")
7656
add_to_lists(c6_1, graph_objects, all_graphs)
7657
7658
# Antiholes
7659
"""
7660
An odd antihole with 7 vertices
7661
p.175
7662
Rebennack, Steffen, Gerhard Reinelt, and Panos M. Pardalos. "A tutorial on branch and cut algorithms for the maximum stable set problem." International Transactions in Operational Research 19.1-2 (2012): 161-199.
7663
7664
sage: antihole_7.order()
7665
7
7666
sage: antihole_7.size()
7667
14
7668
"""
7669
antihole_7 = Antihole(7)
7670
antihole_7.name(new = "Antihole 7")
7671
add_to_lists(antihole_7, graph_objects, all_graphs)
7672
7673
antihole_8 = Antihole(8)
7674
antihole_8.name(new = "Antihole 8")
7675
add_to_lists(antihole_8, graph_objects, all_graphs)
7676
7677
"""
7678
p.10
7679
Barrus, Michael D. "On fractional realizations of graph degree sequences." arXiv preprint arXiv:1310.1112 (2013).
7680
"""
7681
fish = Graph("E@ro")
7682
fish.name(new = "fish")
7683
add_to_lists(fish, graph_objects, all_graphs)
7684
7685
"""
7686
p.10
7687
Barrus, Michael D. "On fractional realizations of graph degree sequences." arXiv preprint arXiv:1310.1112 (2013).
7688
"""
7689
fish_mod = Graph("EiKw")
7690
fish_mod.name(new = "fish_mod")
7691
add_to_lists(fish_mod, graph_objects, all_graphs)
7692
7693
"""
7694
All of the graphs with the name barrus_[0-9]{6} comes from the appendix of
7695
Barrus, Michael D. "On fractional realizations of graph degree sequences." arXiv preprint arXiv:1310.1112 (2013).
7696
"""
7697
7698
barrus_322111a = Graph("EAIW")
7699
barrus_322111a.name(new = "barrus_322111a")
7700
add_to_lists(barrus_322111a, graph_objects, all_graphs)
7701
7702
barrus_322111b = Graph("E?NO")
7703
barrus_322111b.name(new = "barrus_322111b")
7704
add_to_lists(barrus_322111b, graph_objects, all_graphs)
7705
7706
barrus_322221b = Graph("ECXo")
7707
barrus_322221b.name(new = "barrus_322221b")
7708
add_to_lists(barrus_322221b, graph_objects, all_graphs)
7709
7710
barrus_322221c = Graph("EAN_")
7711
barrus_322221c.name(new = "barrus_322221c")
7712
add_to_lists(barrus_322221c, graph_objects, all_graphs)
7713
7714
barrus_332211c = Graph("E@hW")
7715
barrus_332211c.name(new = "barrus_332211c")
7716
add_to_lists(barrus_332211c, graph_objects, all_graphs)
7717
7718
barrus_332211d = Graph("E?lo")
7719
barrus_332211d.name(new = "barrus_332211d")
7720
add_to_lists(barrus_332211d, graph_objects, all_graphs)
7721
7722
barrus_332222a = Graph("E`ow")
7723
barrus_332222a.name(new = "barrus_332222a")
7724
add_to_lists(barrus_332222a, graph_objects, all_graphs)
7725
7726
barrus_332222b = Graph("E`dg")
7727
barrus_332222b.name(new = "barrus_332222b")
7728
add_to_lists(barrus_332222b, graph_objects, all_graphs)
7729
7730
barrus_332222c = Graph("EoSw")
7731
barrus_332222c.name(new = "barrus_332222c")
7732
add_to_lists(barrus_332222c, graph_objects, all_graphs)
7733
7734
barrus_332222d = Graph("E_lo")
7735
barrus_332222d.name(new = "barrus_332222d")
7736
add_to_lists(barrus_332222d, graph_objects, all_graphs)
7737
7738
barrus_333221a = Graph("E`LW")
7739
barrus_333221a.name(new = "barrus_333221a")
7740
add_to_lists(barrus_333221a, graph_objects, all_graphs)
7741
7742
# CE to claw-free & chordal => Hamiltonian
7743
barrus_333221b = Graph("EKSw")
7744
barrus_333221b.name(new = "barrus_333221b(c5_chord_tail)")
7745
add_to_lists(barrus_333221b, graph_objects, all_graphs)
7746
c5_chord_tail = barrus_333221b
7747
add_to_lists(c5_chord_tail, graph_objects, all_graphs)
7748
7749
barrus_333221c = Graph("EELg")
7750
barrus_333221c.name(new = "barrus_333221c")
7751
add_to_lists(barrus_333221c, graph_objects, all_graphs)
7752
7753
barrus_333221d = Graph("EC\o")
7754
barrus_333221d.name(new = "barrus_333221d")
7755
add_to_lists(barrus_333221d, graph_objects, all_graphs)
7756
7757
barrus_333311 = Graph("E@lo")
7758
barrus_333311.name(new = "barrus_333311")
7759
add_to_lists(barrus_333311, graph_objects, all_graphs)
7760
7761
barrus_333322a = Graph("ES\o")
7762
barrus_333322a.name(new = "barrus_333322a")
7763
add_to_lists(barrus_333322a, graph_objects, all_graphs)
7764
7765
barrus_333322c = Graph("ED^_")
7766
barrus_333322c.name(new = "barrus_333322c")
7767
add_to_lists(barrus_333322c, graph_objects, all_graphs)
7768
7769
barrus_422211a = Graph("EGEw")
7770
barrus_422211a.name(new = "barrus_422211a")
7771
add_to_lists(barrus_422211a, graph_objects, all_graphs)
7772
7773
barrus_422211b = Graph("E?No")
7774
barrus_422211b.name(new = "barrus_422211b")
7775
add_to_lists(barrus_422211b, graph_objects, all_graphs)
7776
7777
barrus_432221a = Graph("E@pw")
7778
barrus_432221a.name(new = "barrus_432221a")
7779
add_to_lists(barrus_432221a, graph_objects, all_graphs)
7780
7781
barrus_432221b = Graph("E_Lw")
7782
barrus_432221b.name(new = "barrus_432221b")
7783
add_to_lists(barrus_432221b, graph_objects, all_graphs)
7784
7785
barrus_432221c = Graph("EANg")
7786
barrus_432221c.name(new = "barrus_432221c")
7787
add_to_lists(barrus_432221c, graph_objects, all_graphs)
7788
7789
barrus_432221d = Graph("E?^o")
7790
barrus_432221d.name(new = "barrus_432221d")
7791
add_to_lists(barrus_432221d, graph_objects, all_graphs)
7792
7793
barrus_433222a = Graph("E@vo")
7794
barrus_433222a.name(new = "barrus_433222a")
7795
add_to_lists(barrus_433222a, graph_objects, all_graphs)
7796
7797
barrus_433222c = Graph("EPVW")
7798
barrus_433222c.name(new = "barrus_433222c")
7799
add_to_lists(barrus_433222c, graph_objects, all_graphs)
7800
7801
barrus_433222d = Graph("E`NW")
7802
barrus_433222d.name(new = "barrus_433222d")
7803
add_to_lists(barrus_433222d, graph_objects, all_graphs)
7804
7805
barrus_433321a = Graph("EIMw")
7806
barrus_433321a.name(new = "barrus_433321a")
7807
add_to_lists(barrus_433321a, graph_objects, all_graphs)
7808
7809
barrus_433321b = Graph("E@^o")
7810
barrus_433321b.name(new = "barrus_433321b")
7811
add_to_lists(barrus_433321b, graph_objects, all_graphs)
7812
7813
barrus_433321c = Graph("EPTw")
7814
barrus_433321c.name(new = "barrus_433321c")
7815
add_to_lists(barrus_433321c, graph_objects, all_graphs)
7816
7817
barrus_433332a = Graph("EBzo")
7818
barrus_433332a.name(new = "barrus_433332a")
7819
add_to_lists(barrus_433332a, graph_objects, all_graphs)
7820
7821
barrus_433332b = Graph("E`^o")
7822
barrus_433332b.name(new = "barrus_433332b")
7823
add_to_lists(barrus_433332b, graph_objects, all_graphs)
7824
7825
barrus_433332c = Graph("EqLw")
7826
barrus_433332c.name(new = "barrus_433332c")
7827
add_to_lists(barrus_433332c, graph_objects, all_graphs)
7828
7829
barrus_442222a = Graph("E?~o")
7830
barrus_442222a.name(new = "barrus_442222a")
7831
add_to_lists(barrus_442222a, graph_objects, all_graphs)
7832
7833
barrus_442222b = Graph("E_lw")
7834
barrus_442222b.name(new = "barrus_442222b")
7835
add_to_lists(barrus_442222b, graph_objects, all_graphs)
7836
7837
barrus_443322a = Graph("E@~o")
7838
barrus_443322a.name(new = "barrus_443322a")
7839
add_to_lists(barrus_443322a, graph_objects, all_graphs)
7840
7841
barrus_443322b = Graph("EHuw")
7842
barrus_443322b.name(new = "barrus_443322b")
7843
add_to_lists(barrus_443322b, graph_objects, all_graphs)
7844
7845
barrus_443322c = Graph("EImw")
7846
barrus_443322c.name(new = "barrus_443322c")
7847
add_to_lists(barrus_443322c, graph_objects, all_graphs)
7848
7849
barrus_443322d = Graph("EQlw")
7850
barrus_443322d.name(new = "barrus_443322d")
7851
add_to_lists(barrus_443322d, graph_objects, all_graphs)
7852
7853
barrus_443322e = Graph("E`lw")
7854
barrus_443322e.name(new = "barrus_443322e")
7855
add_to_lists(barrus_443322e, graph_objects, all_graphs)
7856
7857
barrus_443331a = Graph("EBxw")
7858
barrus_443331a.name(new = "barrus_443331a")
7859
add_to_lists(barrus_443331a, graph_objects, all_graphs)
7860
7861
barrus_443331b = Graph("E`\w")
7862
barrus_443331b.name(new = "barrus_443331b")
7863
add_to_lists(barrus_443331b, graph_objects, all_graphs)
7864
7865
barrus_443333a = Graph("Es\w")
7866
barrus_443333a.name(new = "barrus_443333a")
7867
add_to_lists(barrus_443333a, graph_objects, all_graphs)
7868
7869
barrus_443333c = Graph("Eqlw")
7870
barrus_443333c.name(new = "barrus_443333c")
7871
add_to_lists(barrus_443333c, graph_objects, all_graphs)
7872
7873
barrus_444332b = Graph("Ed\w")
7874
barrus_444332b.name(new = "barrus_444332b")
7875
add_to_lists(barrus_444332b, graph_objects, all_graphs)
7876
7877
barrus_444332c = Graph("EMlw")
7878
barrus_444332c.name(new = "barrus_444332c")
7879
add_to_lists(barrus_444332c, graph_objects, all_graphs)
7880
7881
barrus_444433a = Graph("ER~o")
7882
barrus_444433a.name(new = "barrus_444433a")
7883
add_to_lists(barrus_444433a, graph_objects, all_graphs)
7884
7885
barrus_444433b = Graph("Et\w")
7886
barrus_444433b.name(new = "barrus_444433b")
7887
add_to_lists(barrus_444433b, graph_objects, all_graphs)
7888
7889
"""
7890
From:
7891
Alcón, Liliana, Marisa Gutierrez, and Glenn Hurlbert. "Pebbling in split graphs." SIAM Journal on Discrete Mathematics 28.3 (2014): 1449-1466.
7892
"""
7893
pyramid = Graph("EElw")
7894
pyramid.name(new = "pyramid")
7895
add_to_lists(pyramid, graph_objects, all_graphs)
7896
7897
"""
7898
From:
7899
p.3
7900
Dvořák, Zdeněk, and Jordan Venters. "Triangle-free planar graphs with small independence number." arXiv preprint arXiv:1702.02888 (2017).
7901
"""
7902
thomas_walls_3 = Graph("IRaIACbFG")
7903
thomas_walls_3.name(new = "thomas_walls_3")
7904
add_to_lists(thomas_walls_3, graph_objects, all_graphs)
7905
7906
"""
7907
From:
7908
p.3
7909
Dvořák, Zdeněk, and Jordan Venters. "Triangle-free planar graphs with small independence number." arXiv preprint arXiv:1702.02888 (2017).
7910
"""
7911
thomas_walls_4 = Graph("LQCkCD?OGM@EKB")
7912
thomas_walls_4.name(new = "thomas_walls_4")
7913
add_to_lists(thomas_walls_4, graph_objects, all_graphs)
7914
7915
"""
7916
P. 24, 1st figure
7917
From a knuth paper, linked in issue #342
7918
"""
7919
knuth_24_1 = Graph("E`~w")
7920
knuth_24_1.name(new = "knuth_24_1")
7921
add_to_lists(knuth_24_1, graph_objects, all_graphs)
7922
7923
"""
7924
P. 24, 2nd figure
7925
From a knuth paper, linked in issue #343
7926
"""
7927
knuth_24_2 = Graph("Er~w")
7928
knuth_24_2.name(new = "knuth_24_2")
7929
add_to_lists(knuth_24_2, graph_objects, all_graphs)
7930
7931
"""
7932
From Craig Larson, Critical Independence paper
7933
Has non empty critical independence set
7934
"""
7935
larson = Graph("F@N~w")
7936
larson.name(new = "larson")
7937
add_to_lists(larson, graph_objects, all_graphs)
7938
7939
"""
7940
From:
7941
Vizing's independence number conjecture is true asymptotically
7942
Eckhard Steffen
7943
"""
7944
steffen_1 = Graph("HqP@xw{")
7945
steffen_1.name(new = "steffen_1")
7946
add_to_lists(steffen_1, graph_objects, all_graphs)
7947
7948
"""
7949
From:
7950
Vizing's independence number conjecture is true asymptotically
7951
Eckhard Steffen
7952
"""
7953
steffen_2 = Graph("N??xpow@o@?AoBoBW@_")
7954
steffen_2.name(new = "steffen_2")
7955
add_to_lists(steffen_2, graph_objects, all_graphs)
7956
7957
"""
7958
From:
7959
Vizing's independence number conjecture is true asymptotically
7960
Eckhard Steffen
7961
"""
7962
steffen_3 = Graph("Z??xpow@o@?A?B?B?@_?C??_?@??@??A???__?Dc??kO?AoO?AWO?AWG?@K?")
7963
steffen_3.name(new = "steffen_3")
7964
add_to_lists(steffen_3, graph_objects, all_graphs)
7965
7966
"""
7967
From:
7968
p.231
7969
Balinski, Michel L. "Notes—On a Selection Problem." Management Science 17.3 (1970): 230-231.
7970
"""
7971
balanski = Graph("H??xuRo")
7972
balanski.name(new = "balanski")
7973
add_to_lists(balanski, graph_objects, all_graphs)
7974
7975
"""
7976
Instance of the Erdos-Faber-Lovasz conjecture
7977
"""
7978
efl_instance = Graph("J`?GECrB~z_")
7979
efl_instance.name(new = "efl_instance")
7980
add_to_lists(efl_instance, graph_objects, all_graphs)
7981
7982
"""
7983
From Lipták, László, and László Lovász. "Critical facets of the stable set polytope." Combinatorica 21.1 (2001): 61-88.
7984
with defect 2
7985
"""
7986
basic_facet_1 = Graph("FCS~?")
7987
basic_facet_1.name(new = "basic facet-graph")
7988
add_to_lists(basic_facet_1, graph_objects, all_graphs)
7989
7990
"""
7991
From Lipták, László, and László Lovász. "Critical facets of the stable set polytope." Combinatorica 21.1 (2001): 61-88.
7992
with defect 2
7993
"""
7994
basic_facet_2 = Graph("I_?BSgcEG")
7995
basic_facet_2.name(new = "basic facet-graph two")
7996
add_to_lists(basic_facet_2, graph_objects, all_graphs)
7997
7998
"""
7999
From Lipták, László, and László Lovász. "Critical facets of the stable set polytope." Combinatorica 21.1 (2001): 61-88.
8000
with defect 2
8001
"""
8002
basic_facet_3 = Graph("L@Q????o`QAc?w")
8003
basic_facet_3.name(new = "basic facet-graph three")
8004
add_to_lists(basic_facet_3, graph_objects, all_graphs)
8005
8006
"""
8007
From Lipták, László, and László Lovász. "Critical facets of the stable set polytope." Combinatorica 21.1 (2001): 61-88.
8008
"""
8009
crit_facet_1 = Graph("H@BDQo{")
8010
crit_facet_1.name(new = "critical facet graph 1")
8011
add_to_lists(crit_facet_1, graph_objects, all_graphs)
8012
8013
"""
8014
From Lipták, László, and László Lovász. "Critical facets of the stable set polytope." Combinatorica 21.1 (2001): 61-88.
8015
"""
8016
crit_facet_2 = Graph("J?AAHGIC^o?")
8017
crit_facet_2.name(new = "critical facet graph 2")
8018
add_to_lists(crit_facet_2, graph_objects, all_graphs)
8019
8020
"""
8021
From:
8022
p.222
8023
Tarjan, Robert E. "Decomposition by clique separators." Discrete mathematics 55.2 (1985): 221-232.
8024
"""
8025
tarjan = Graph("JQ?CQOKDXL_")
8026
tarjan.name(new = "tarjan")
8027
add_to_lists(tarjan, graph_objects, all_graphs)
8028
8029
"""
8030
From:
8031
p.225
8032
Tarjan, Robert E. "Decomposition by clique separators." Discrete mathematics 55.2 (1985): 221-232.
8033
"""
8034
tarjan_fillin = Graph("JQAa_OFJWv_")
8035
tarjan_fillin.name(new = "tarjan_fillin")
8036
add_to_lists(tarjan_fillin, graph_objects, all_graphs)
8037
8038
"""
8039
From:
8040
p.252
8041
Sewell, E. C., and L. E. Trotter. "Stability critical graphs and ranks facets of the stable set polytope." Discrete Mathematics 147.1-3 (1995): 247-255.
8042
"""
8043
sewell_1 = Graph("H@GOkXI")
8044
sewell_1.name(new = "sewell_1")
8045
add_to_lists(sewell_1, graph_objects, all_graphs)
8046
8047
"""
8048
Graphs which are two cliques somehow connected to each other
8049
"""
8050
two_clique_1 = Graph("OwCW?CB???_Bw?F?_[W?~")
8051
two_clique_1.name(new = "two-clique 1")
8052
add_to_lists(two_clique_1, graph_objects, all_graphs)
8053
8054
two_clique_2 = Graph("FwC^w")
8055
two_clique_2.name(new = "two-clique 2")
8056
add_to_lists(two_clique_2, graph_objects, all_graphs)
8057
8058
caro_roditty_4 = Caro_Roditty(4)
8059
caro_roditty_4.name(new = "Caro-Roditty 4 Graph")
8060
add_to_lists(caro_roditty_4, graph_objects, all_graphs)
8061
8062
caro_roditty_5 = Caro_Roditty(5)
8063
caro_roditty_5.name(new = "Caro-Roditty 5 Graph")
8064
add_to_lists(caro_roditty_5, graph_objects, all_graphs)
8065
8066
jakovac_2a = Graph("a???????GO???@???_?????C??W????????@_A???B??W??[??????X?E?Q?????????C??Bb???C?o?K??s?@?GZ??F?Ew")
8067
jakovac_2a.name(new = "jakovac fig 2a")
8068
add_to_lists(jakovac_2a, graph_objects, all_graphs)
8069
8070
jakovac_2b = Graph("a???????????????????????@???C??????C?????oO??_I?G?o@_??[????o?O`?????????G???AW???_[??O_N??W?Bw")
8071
jakovac_2b.name(new = "jakovac fig 2b")
8072
add_to_lists(jakovac_2b, graph_objects, all_graphs)
8073
8074
jakovac_5 = Graph("Q`?G?C??G??@?B?K?W?W@o?go@O")
8075
jakovac_5.name(new = "jakovac fig 5")
8076
add_to_lists(jakovac_5, graph_objects, all_graphs)
8077
8078
# From A survey of selected recent results on total domination in graphs By Michael A. Henning
8079
henning_5 = Graph("H?qipno")
8080
henning_5.name(new = "henning fig 5")
8081
add_to_lists(henning_5, graph_objects, all_graphs)
8082
8083
henning_8 = Graph("[SP???_C?O?_?_?O?G?C????????CO?AG??c??E??W??E?????@???A_??QG??BD")
8084
henning_8.name(new = "henning fig 8")
8085
add_to_lists(henning_8, graph_objects, all_graphs)
8086
8087
henning_13 = Graph("khCGGC@?G?_@?@??_?G?@??C??G??K??EG?????KO??_??@???@???G__??????@?C??C????G????G???@C?G_???????G?@???_????@?????@?????G_?A?????????@??CO??C??????G??????G?????@C")
8088
henning_13.name(new = "henning fig 13")
8089
add_to_lists(henning_13, graph_objects, all_graphs)
8090
8091
henning_14 = Graph("M_GSA?_C?G@??b?[?")
8092
henning_14.name(new = "henning fig 14")
8093
add_to_lists(henning_14, graph_objects, all_graphs)
8094
8095
henning_15 = graphs.CycleGraph(4)
8096
for v in henning_15.vertices():
8097
count = len(henning_15.vertices())
8098
henning_15.add_cycle(range(count,count+10))
8099
henning_15.add_edge((v, count))
8100
henning_15.name(new = "henning fig 15")
8101
add_to_lists(henning_15, graph_objects, all_graphs)
8102
8103
henning_16_g1 = Graph("Fq?gw")
8104
henning_16_g1.name(new = "henning fig 16 g1")
8105
add_to_lists(henning_16_g1, graph_objects, all_graphs)
8106
8107
henning_16_g2 = Graph("FWEYo")
8108
henning_16_g2.name(new = "henning fig 16 g2")
8109
add_to_lists(henning_16_g2, graph_objects, all_graphs)
8110
8111
henning_16_g3 = Graph("JQQ@?cCAGF_")
8112
henning_16_g3.name(new = "henning fig 16 g3")
8113
add_to_lists(henning_16_g3, graph_objects, all_graphs)
8114
8115
henning_16_g4 = Graph("NC`@A?_C?@_JA??___W")
8116
henning_16_g4.name(new = "henning fig 16 g4")
8117
add_to_lists(henning_16_g4, graph_objects, all_graphs)
8118
8119
henning_16_g5 = Graph("JSP@?cG@GF_")
8120
henning_16_g5.name(new = "henning fig 16 g5")
8121
add_to_lists(henning_16_g5, graph_objects, all_graphs)
8122
8123
henning_16_g6 = Graph("LSP??CAGOa`C@H")
8124
henning_16_g6.name(new = "henning fig 16 g6")
8125
add_to_lists(henning_16_g6, graph_objects, all_graphs)
8126
8127
henning_16_g7 = Graph("NCGaC@?C?@_J?_A?__W")
8128
henning_16_g7.name(new = "henning fig 16 g7")
8129
add_to_lists(henning_16_g7, graph_objects, all_graphs)
8130
8131
"""
8132
From:
8133
Weighted and Unweighted Maximum Clique Algorithms with Upper Bounds from Fractional Coloring
8134
E. Balas and Jue Xue
8135
"""
8136
balas_1 = Graph("HWFZvq}")
8137
balas_1.name(new = "balas_1")
8138
add_to_lists(balas_1, graph_objects, all_graphs)
8139
8140
"""
8141
From:
8142
On alpha+ - Stable K¨onig-Egervary Graphs
8143
Vadim E. Levit and Eugen Mandrescu
8144
"""
8145
levit_1 = Graph("G?rHx{")
8146
levit_1.name(new = "levit_1")
8147
add_to_lists(levit_1, graph_objects, all_graphs)
8148
8149
"""
8150
From:
8151
On alpha+ - Stable K¨onig-Egervary Graphs
8152
Vadim E. Levit and Eugen Mandrescu
8153
"""
8154
levit_2 = Graph("GCOihs")
8155
levit_2.name(new = "levit_2")
8156
add_to_lists(levit_2, graph_objects, all_graphs)
8157
8158
"""
8159
From:
8160
On alpha+ - Stable K¨onig-Egervary Graphs
8161
Vadim E. Levit and Eugen Mandrescu
8162
"""
8163
levit_3 = Graph("G@G]@{")
8164
levit_3.name(new = "levit_3")
8165
add_to_lists(levit_3, graph_objects, all_graphs)
8166
8167
"""
8168
From:
8169
On alpha-Critical Edges in K¨onig-Egerv´ary Graphs
8170
Vadim E. Levit and Eugen Mandrescu
8171
"""
8172
levit_4 = Graph("KGD?_O_@GE~~")
8173
levit_4.name(new = "levit_4")
8174
add_to_lists(levit_4, graph_objects, all_graphs)
8175
8176
# Two cycles of order cycle1 and cycle2 adjoined at a single vertex.
8177
def glasses_graph(cycle1, cycle2):
8178
glasses = graphs.CycleGraph(cycle1)
8179
glasses.add_cycle(range(cycle1, cycle1 + cycle2))
8180
glasses.merge_vertices([0, cycle1])
8181
return glasses
8182
8183
# CE to (~weakly_chordial) & is_eularian => is_hamiltonian
8184
glasses_5_5 = glasses_graph(5,5)
8185
add_to_lists(glasses_5_5, graph_objects, all_graphs)
8186
8187
# CE to Hamiltonian => g.is_bipartite() and g.is_strongly_regular()
8188
# CE to Hamiltonian => g.is_circular_planar() and g.is_bipartite()
8189
ham1 = Graph("I~EGYCxyG")
8190
ham1.name(new = "ham1")
8191
add_to_lists(ham1, graph_objects, all_graphs)
8192
8193
# CE to ((is_van_den_heuvel)&(is_circular_planar))->(is_hamiltonian)
8194
umbrella_4 = Graph("Ep{G")
8195
umbrella_4.name(new = "umbrella_4")
8196
add_to_lists(umbrella_4, graph_objects, all_graphs)
8197
8198
def kite_necklace(k):
8199
#returns k kites joined head to tail
8200
if k == 1:
8201
return Graph('DJk')
8202
g = graphs.CycleGraph(2*k)
8203
for i in range(k):
8204
g.subdivide_edge((2*i, 1 + 2*i), 1)
8205
g.add_edge(2*i, 1 + 2*i)
8206
g.subdivide_edge((2*i, 1 + 2*i), 1)
8207
g.add_edge(2*k + 2*i, 2*k + 2*i + 1)
8208
return g
8209
8210
# CE to (is_hamiltonian)->( (is_regular)->(is_kite_free) )
8211
kite_necklace_3 = kite_necklace(3)
8212
kite_necklace_3.name(new = "kite_necklace_3")
8213
add_to_lists(kite_necklace_3, graph_objects, all_graphs)
8214
8215
# CE to (locally connected)->(is_hamiltonian)
8216
bus = graphs.CycleGraph(4)
8217
bus.add_vertex(4)
8218
bus.add_edges([(0,4),(2,4),(0,2)])
8219
bus.name(new = "bus")
8220
add_to_lists(bus, graph_objects, all_graphs)
8221
8222
# from p.37 of Algorithmic Graph Theory In Sage by authors DaMNeD
8223
DaMNeD1 = Graph('E@~w')
8224
DaMNeD1.name(new = "DaMNeD1")
8225
DaMNeD2 = Graph('F@N]w')
8226
DaMNeD2.name(new = "DaMNeD2")
8227
DaMNeD3 = Graph('GIaHx{')
8228
DaMNeD3.name(new = "DaMNeD3")
8229
DaMNeD4 = Graph('HIaJ`{~')
8230
DaMNeD4.name(new = "DaMNeD4")
8231
add_to_lists(DaMNeD1, graph_objects, all_graphs)
8232
add_to_lists(DaMNeD2, graph_objects, all_graphs)
8233
add_to_lists(DaMNeD3, graph_objects, all_graphs)
8234
add_to_lists(DaMNeD4, graph_objects, all_graphs)
8235
8236
# These graphs are cubic, planar, and vertex transitive
8237
tricone = Graph('It?IQGiDO')
8238
tricone.name(new = "tricone")
8239
truncatedCubical = Graph('Wt?GWGAA?A?C?C?C_?_?O??S?A_??o?@H??@O??gaO?CC_?')
8240
truncatedCubical.name(new = "truncatedCubical")
8241
truncatedDodecahedral = Graph('{t?GW???oH?G?G??_?gO??S??g??E??H???O??CG??A???O??@???A????A???@?????O???A?????g???@O???C????A?????AO????O_????GO????C_??????_?????O??????O?????A??????__????@@???????C??????AG??????c??????CG??????@G??????AC????????_???????O????????c???????@_????????o???????@H????????@O????????g_??G@????C??@?A????')
8242
truncatedDodecahedral.name(new = "truncatedDodecahedral")
8243
truncatedOctahedral = Graph ('Ws?GO?@?O??????D?CO@o?_GC@?M??_W?OI?G@@A?COL???')
8244
truncatedOctahedral.name(new = "truncatedOctahedral")
8245
add_to_lists(tricone, graph_objects, all_graphs)
8246
add_to_lists(truncatedCubical, graph_objects, all_graphs)
8247
add_to_lists(truncatedDodecahedral, graph_objects, all_graphs)
8248
add_to_lists(truncatedOctahedral, graph_objects, all_graphs)
8249
8250
# CE to ((is_heliotropic_plant)&(is_regular))->(is_hamiltonian)
8251
crossingNumberGraph3E = Graph('Ms??WX?CH@CQCQR??')
8252
crossingNumberGraph3E.name(new = "crossingNumberGraph3E")
8253
add_to_lists(crossingNumberGraph3E, graph_objects, all_graphs)
8254
8255
# CE to ((is_quasi_regular)&(is_cartesian_product))->(is_hamiltonian)
8256
ce136 = Graph('K{CY?CBG??_B').cartesian_product(graphs.CompleteGraph(2))
8257
ce136.name(new = "ce136")
8258
add_to_lists(ce136, graph_objects, all_graphs)
8259
8260
# CE to (((has_radius_equal_diameter)&(is_line_graph))&(~(is_perfect)))->(is_hamiltonian)
8261
# CE to (((has_radius_equal_diameter)&(is_line_graph))&(~(is_quasi_regular)))->(is_hamiltonian)
8262
# (equivalent) CE to (((diameter_equals_radius)&(is_line_graph))&(~(is_quasi_regular)))->(is_hamiltonian)
8263
ce137 = Graph('X~aKWOHAYA?H?ROAG@H@?G@C_?WGGOGACC?B?GSO@?__?`O??OS')
8264
ce137.name(new = "ce137")
8265
add_to_lists(ce137, graph_objects, all_graphs)
8266
8267
# CE to ((~(is_perfect))&(girth_greater_than_2log))->(is_hamiltonian)
8268
glasses_17_17 = glasses_graph(17,17)
8269
glasses_17_17.name(new = "glasses_17_17")
8270
add_to_lists(glasses_17_17, graph_objects, all_graphs)
8271
8272
# CE to (((is_factor_critical)&(has_radius_equal_diameter))&((is_perfect)^(has_lovasz_theta_equals_alpha)))->(is_hamiltonian)
8273
ce138 = Graph('N_goPePG?_POB_aWyB_')
8274
ce138.name(new = "ce138")
8275
add_to_lists(ce138, graph_objects, all_graphs)
8276
8277
# CE to (((is_quasi_regular)&(is_H_free))&(has_dart))->(is_hamiltonian)
8278
ce139 = Graph('Iv??G[N]?')
8279
ce139.name(new = "ce139")
8280
add_to_lists(ce139, graph_objects, all_graphs)
8281
8282
# CE to (((has_radius_equal_diameter)&(has_kite))&(is_quasi_regular))->(is_hamiltonian)
8283
ce140 = Graph('Mt?GOGB@?CgHOKM??')
8284
ce140.name(new = "ce140")
8285
add_to_lists(ce140, graph_objects, all_graphs)
8286
8287
# CE to ((~((is_quasi_regular)|(is_perfect)))&(has_radius_equal_diameter))->(is_hamiltonian)
8288
ce141 = Graph('KGh?Q?ALCjQ[')
8289
ce141.name(new = "ce141")
8290
add_to_lists(ce141, graph_objects, all_graphs)
8291
8292
# CE to ((~(is_c4_free))&(is_locally_connected))->(is_hamiltonian)
8293
ce142 = Graph('F?~vw')
8294
ce142.name(new = "ce142")
8295
add_to_lists(ce142, graph_objects, all_graphs)
8296
8297
# CE to (is_perfect & is_three_connected & alpha_leq_order_over_two)->(is_hamiltonian)
8298
# CE to (((is_factor_critical)^(is_independence_irreducible))&((is_H_free)&(has_radius_equal_diameter)))->(is_hamiltonian)
8299
triangleReplacedPetersen = Graph(']t?G?C????_CAG@C?H?CG?G_?P??G_?P???G???W??G???P???K???BG???I???DF????b????')
8300
triangleReplacedPetersen.name(new = "triangleReplacedPetersen")
8301
add_to_lists(triangleReplacedPetersen, graph_objects, all_graphs)
8302
8303
# CE to ((alpha_leq_order_over_two)&(is_locally_two_connected))->(is_hamiltonian)
8304
ce143 = Graph('K~e[Mm@oYaAg')
8305
ce143.name(new = "ce143")
8306
add_to_lists(ce143, graph_objects, all_graphs)
8307
8308
# CE to (is_hamiltonian)->((order_leq_twice_max_degree)|(is_locally_bipartite))
8309
ce144 = Graph('LdEIDOHA{dc_Sa')
8310
ce144.name(new = "ce144")
8311
add_to_lists(ce144, graph_objects, all_graphs)
8312
8313
petersen_line_graph = (graphs.PetersenGraph()).line_graph()
8314
petersen_line_graph.relabel()
8315
petersen_line_graph.name(new = "Line graph of Petersen graph")
8316
add_to_lists(petersen_line_graph, graph_objects, all_graphs)
8317
8318
# CE to is_locally_unicyclic -> is_hamiltonian
8319
ce145 = Graph('K~?GW[PeW@_F')
8320
ce145.name(new = "ce145") # Contributed by Sarah
8321
add_to_lists(ce145, graph_objects, all_graphs)
8322
8323
# CE to independence number >= min( girth, floor(lovasz_theta) )
8324
ce146 = Graph('hUYuvZu}rzVuvuzz]}vvu^^Y}}q}}t^^ZVvuy}}vjzzNVvunVvuVjzzTy}}unVvuy|^^Zty}}vty}}ry|^^Y}nVvuvty}}v^Vjzz]}nVvu}}nVvu^^VjzzVvty}}u}}nVvuZzy|^^W')
8325
ce146.name(new = "ce146") # Contributed by Jianxian Chen, @a234
8326
add_to_lists(ce146, graph_objects, all_graphs)
8327
8328
c4_free_subgraph_of_Q3 = Graph('Gb_k@C')
8329
c4_free_subgraph_of_Q3.name(new = "c4_free_subgraph_of_Q3")
8330
add_to_lists(c4_free_subgraph_of_Q3, graph_objects, all_graphs)
8331
8332
ce147 = Graph('L`?M@_LBf_Q~Y}')
8333
ce147.name(new = "ce147")
8334
add_to_lists(ce147, graph_objects, all_graphs)
8335
8336
# Vincent Cohen-Addad, Michael Hebdige, Daniel Král', Zhentao Li, and Esteban Salgado, Steinberg's Conjecture is false. Journal of Combinatorial Theory, Series B, 2017.
8337
CaHKLS17 = Graph("~?AeCsAGK_WG??@?B?A?@GG@@?C??I??I?AC?@@???G???_??b??KA?@????_?G@??@?G??C????I????I???AC???@@?????G?????_????b????KA??_?????O???G@????@?G????C??????GG?????GG????AC?????@@???????G???????_??????b??????KA????????????????G???????B????????O???????@I????????I????????C????????@_????????K????????O_???????@@?????????@??????????_????????CW????????KA????????G??????????_?H??????????H??????????C??????????@_??????????K??????????O_?????????@@???????????@????????????_??????????CW??????????KA????????C????????????O???H????????????H????????????C????????????@O????????????I????????????O_???????????@@?????????????@??????????????_????????????CW????????????KA?????????????????????????????G??????????????W??????????????O??????????????H?O?????????????GA??????????????_??????????????@O??????????????@O??????????????O_??????????????GG???????????????@????????????????C???????????????CW??????????????@_O??????????????G????????????????C?@?G???????????????G@????????????????_????????????????@O????????????????@O????????????????O_????????????????GG?????????????????@??????????????????C?????????????????CW????????????????@_O??????????????C??????????????????A???@?G?????????????????G@??????????????????_??????????????????@?O?????????????????@?O?????????????????O_??????????????????GG???????????????????@????????????????????C???????????????????CW??????????????????@_O?????????????????????????????????????????@?????????????????????W????????????????????A?????????????????????HG????????????????????@G?????????????????????_?????????????????????I?????????????????????@O????????????????????AC?????????????????????GG??????????????????????G??????????????????????C??????????????????????b?????????????????????@_O????????????????????@???????????????????????C?@C??????????????????????@C???????????????????????_???????????????????????I???????????????????????@O??????????????????????AC???????????????????????GG????????????????????????G????????????????????????C????????????????????????b???????????????????????@_O?????????????????????_????????????????????????A???@C????????????????????????@C?????????????????????????_?????????????????????????H?????????????????????????@G????????????????????????AC?????????????????????????GG??????????????????????????G??????????????????????????C??????????????????????????b?????????????????????????@_O")
8338
CaHKLS17.name(new = "CaHKLS17")
8339
add_to_lists(CaHKLS17, graph_objects, all_graphs)
8340
8341
# Dmytrenko, V. , Lazebnik, F. and Viglione, R. (2005), Isomorphism criterion for monomial graphs. J. Graph Theory, 48: 322-328. doi:10.1002/jgt.20055
8342
DLV05 = Graph( 'Qs???CA??B_M?_B?`_QW?KC@o??')
8343
DLV05.name(new = "DLV05")
8344
add_to_lists(DLV05, graph_objects, all_graphs)
8345
8346
# Garnick, D. K., Kwong, Y. H. and Lazebnik, F. (1993), Extremal graphs without three‐cycles or four‐cycles. J. Graph Theory, 17: 633-645. doi:10.1002/jgt.3190170511
8347
GKL11 = Graph('Jo?OQoqJAK?')
8348
GKL11.name(new = "GKL11")
8349
add_to_lists(GKL11, graph_objects, all_graphs)
8350
8351
triangleReplacedCoxeter = Graph("~?@S{CWOCB?O?_B?A??_?W?A??C??W??O??C??B@?????A_??B???@????_???W???A????C????W????O????C????B?????O?????_????B?????A??????_?????Y????????????S??????W??????G??????C??????B????@?????????A_??????B??@????????????@_???????W???????@????????C????????W????????O????????C????????B?????????O?????????_????????B?????????A??????????_?????????W???@????????????????S??????????W??????????G??????????C??????????BG???????????????????G??_??????????J???G??????????????G?????_?????????_?W?@???????????????@??????C???????????_W?G?????????????????@?????C??????????_?B?G????????????????@????????_????????C???B")
8352
triangleReplacedCoxeter.name(new = "triangleReplacedCoxeter")
8353
add_to_lists(triangleReplacedCoxeter, graph_objects, all_graphs)
8354
8355
# Fig 5.4 of Chartrand, Gary, Linda Lesniak, and Ping Zhang. Graphs & digraphs. Vol. 39. CRC press, 2010.
8356
CGLZ10_f54 = Graph('I?_aCwvUw')
8357
CGLZ10_f54.name(new = "CGLZ10_f54")
8358
add_to_lists(CGLZ10_f54, graph_objects, all_graphs)
8359
8360
# Fig 5.5 of Chartrand, Gary, Linda Lesniak, and Ping Zhang. Graphs & digraphs. Vol. 39. CRC press, 2010.
8361
CGLZ10_f55 = Graph('K??H?C@?W`z_')
8362
CGLZ10_f55.name(new = "CGLZ10_f55")
8363
add_to_lists(CGLZ10_f55, graph_objects, all_graphs)
8364
8365
# Jan Goedgebeur, On minimal triangle-free 6-chromatic graphs. 2017. https://arxiv.org/pdf/1707.07581.pdf
8366
Goedgebeur17 = Graph('U???CA?O?O_aGC?`A@?HDGQOPAA@AGE@?Gc@~_??')
8367
Goedgebeur17.name(new = "Goedgebuer17")
8368
add_to_lists(Goedgebeur17, graph_objects, all_graphs)
8369
8370
# David Garnick, Y.H. Harris Kwong, and Felxi Lazebnik, Algorithmic search for extremal graphs of girth at least five. 1992.
8371
GKL92_1 = Graph('Q??G_GBD?SOAOAGKcHaPGcObe??')
8372
GKL92_1.name(new = "GKL92_1")
8373
add_to_lists(GKL92_1, graph_objects, all_graphs)
8374
GKL92_2 = Graph('WhCGKC@QGC_@CPP?_?HC@a@C?`H?OGQ@D?C@@G?K_P?aO?c')
8375
GKL92_2.name(new = "GKL92_2")
8376
add_to_lists(GKL92_2, graph_objects, all_graphs)
8377
8378
# http://mathworld.wolfram.com/ThomassenGraphs.html
8379
thomassen20 = Graph("ShCKK?DOGC_H?@O?__H?HG??C?G_CGOCO")
8380
thomassen20.name(new = "thomassen20")
8381
add_to_lists(thomassen20, graph_objects, all_graphs)
8382
8383
thomassen32 = Graph("_hCKGCH?G?_@?`??_@G?@??C??K?CG??C??H?C?H???o??@O??CC??@aG???G_?@?CO???CO?G??a????G_C")
8384
thomassen32.name(new = "thomassen32")
8385
add_to_lists(thomassen32, graph_objects, all_graphs)
8386
8387
thomassen34 = Graph("aUW??GE?o????A?B@?o???????O??o?_Y?_?O??H???a??@@????_??_G?CG?O?G?@??C?A??G?A_?G??O?C??C?@???__G")
8388
thomassen34.name(new = "thomassen34")
8389
add_to_lists(thomassen34, graph_objects, all_graphs)
8390
8391
thomassen41 = Graph("hhDGHEHC??_@?@??_OG_???C??G??G??E??@C????O?_??@?C?@?@?C_???_?@?@??_?CA???G?A?@G???????G?@C????G??G??_?A??H?????O???G??_??O??K?????@???O??c")
8392
thomassen41.name(new = "thomassen41")
8393
add_to_lists(thomassen41, graph_objects, all_graphs)
8394
8395
thomassen60 = Graph("{hCGGCDAG?o@O???_?G?@??C??GE?GC?G??H???G?C?o?????O@????_???G???@???_C@???G????g????C???C@???@??????A_????C?????d?????B????C?G???C?@????C?C???@?@G?A???????_???C??G???A??G????G??????A_??@???@???G???@???C???A??O?????WO???????G???????C????????G_???????I????????c?C??????C?C???????G?O??????GO????????B")
8396
thomassen60.name(new = "thomassen60")
8397
add_to_lists(thomassen60, graph_objects, all_graphs)
8398
8399
thomassen94 = Graph("~?@]l_GGD@?G?_@G@??_?G?@C?C??G??GO?C??A???G???_??@???@????_???G??@@?A???????G????G????C???A@?????G?????_???@@?_??????????_?????G?????@??????C??????G??????G?????CC??_???????????G???????_??????@???????@????????_???????G??????@@??O?????C?C??????G???????@G????????C?????C??@????@??????????????_??????O?@?C???????@?O?????O??__????G?????????@??????????????C???O??????G????_????????????????C???????G??@??@????????G???????G??A???????????@????????_??@????????????_???????GG??G????????@??A?????A??????C????????????G???@????????G??????????CGC???@?O??????C???@?O?????????????????????_??@?????????@??????????@??@??????????????_?????@???????G?????????????@?????????C????C????A????????GG??????G??O????_??????_???O??A???@???????o??????C????C???C????")
8400
thomassen94.name(new = "thomassen94")
8401
add_to_lists(thomassen94, graph_objects, all_graphs)
8402
8403
thomassen105 = Graph("~?@hhe?IC@GG?a@?@A?__GG???CA?GC????C@?@?O?????_C?@?G??????_A??G?_??????C?G??G?O???????C?C??@?@?????????_?O??@??_?????????_?G???G?A??????????C??_???G?@???????????CC?????@???????G??_????_??_???????_???@???G???????A????G???G????????_???C????_????????_???G????G????????A???@?????G?????????_???_?????_?????????_??@??????G?????????A???G??????G??????????_??C???????G?????O???????G???????G??C???????A??????O???????G???????A???_???????C??????A???????@????????C???_???????@???????A???????@????????@???C????????A????????O???????G????????A???C?????????_????????O???????G?????????_???_????????@?????????A???????@?????????@????_?????????O?????????A???????@??????????O???C??????????_??????????O???????G??????????_???C??????????G???????????O???????G???????_???????_??????????_???@???????????W???@???????????B????_???????????K???G????????????W??@?????????????W??C?????????????K??G?????????????B??G??????????????W?C??????????????@c@")
8404
thomassen105.name(new = "thomassen105")
8405
add_to_lists(thomassen105, graph_objects, all_graphs)
8406
8407
# R.L. Graham, On edgewise 2-colord graphs with monchromatic triangles and containing no complete hexagon. Journal of Combinatorial Theory, 4(3): 300, 1968.
8408
Graham68 = Graph('GUZ~~{')
8409
Graham68.name(new = "Graham68")
8410
add_to_lists(Graham68, graph_objects, all_graphs)
8411
8412
# Distance Regular Graphs:
8413
split_cayley_hex_2 = Graph("~??~{aSO???GA??????_?_???????G??_?????@???????_?_G??_A?@?C??CC??A?_??OO??A?_??@?O?@?GG?@?G?A?GC?O??O_?_?A?O?_?A@?A??O?_G??A@?O?ABo????O@w???D??]???A??Bo??H???GaO?_??@GQ?G???@h?H???D?BA@?_O_?_?C?c?_C??A?Q?CG??A?C_C?O??C@_?CO???_`?@O???CCG@??_??GAC?OA???a?AG@??C?G_?__O?@?CC?CC@??G?OO?S?C?G??W?AG?G?@??c?OA?G?C?@G?A?o???_?`?C?`???_?")
8414
split_cayley_hex_2.name(new = "split_cayley_hex_2")
8415
add_to_lists(split_cayley_hex_2, graph_objects, all_graphs)
8416
# Bipartite, not strongly regular
8417
distreg_not_stronglyreg1 = Graph("QKUalPTiaitSTTiiIilTSTTUii_")
8418
distreg_not_stronglyreg1.name(new = "distreg_not_stronglyreg1")
8419
distreg_not_stronglyreg2 = Graph("IKUalPTi_")
8420
distreg_not_stronglyreg2.name(new = "distreg_not_stronglyreg2")
8421
distreg_not_stronglyreg3 = Graph("KKUalPTiaitS")
8422
distreg_not_stronglyreg3.name(new = "distreg_not_stronglyreg3")
8423
distreg_not_stronglyreg4 = Graph("SKUalPTiaitSTTiiIilTSTTUiiaiilTTO")
8424
distreg_not_stronglyreg4.name(new = "distreg_not_stronglyreg4")
8425
distreg_not_stronglyreg5 = Graph("MKUalPTiaitSTTii?")
8426
distreg_not_stronglyreg5.name(new = "distreg_not_stronglyreg5")
8427
distreg_not_stronglyreg6 = Graph("OKUalPTiaitSTTiiIilTS")
8428
distreg_not_stronglyreg6.name(new = "distreg_not_stronglyreg6")
8429
graph_objects.extend([distreg_not_stronglyreg1, distreg_not_stronglyreg2, distreg_not_stronglyreg3, distreg_not_stronglyreg4, distreg_not_stronglyreg5,
8430
distreg_not_stronglyreg6])
8431
all_graphs.extend([distreg_not_stronglyreg1, distreg_not_stronglyreg2, distreg_not_stronglyreg3, distreg_not_stronglyreg4, distreg_not_stronglyreg5,
8432
distreg_not_stronglyreg6])
8433
# Not bipartite, not strongly regular
8434
distreg_not_stronglyreg7 = Graph("L~~~~~~~~~~~~~")
8435
distreg_not_stronglyreg7.name(new = "distreg_not_stronglyreg7")
8436
distreg_not_stronglyreg8 = Graph("N~~~~~~~~~~~~~~~~~w")
8437
distreg_not_stronglyreg8.name(new = "distreg_not_stronglyreg8")
8438
distreg_not_stronglyreg9 = Graph("P~~~~~~~~~~~~~~~~~~~~~~{")
8439
distreg_not_stronglyreg9.name(new = "distreg_not_stronglyreg9")
8440
distreg_not_stronglyreg10 = Graph("J~~~~~~~~~_")
8441
distreg_not_stronglyreg10.name(new = "distreg_not_stronglyreg10")
8442
distreg_not_stronglyreg11 = Graph("Q~~~~~~~~~~~~~~~~~~~~~~~~~w")
8443
distreg_not_stronglyreg11.name(new = "distreg_not_stronglyreg11")
8444
distreg_not_stronglyreg12 = Graph("M~~~~~~~~~~~~~~~_")
8445
distreg_not_stronglyreg12.name(new = "distreg_not_stronglyreg12")
8446
distreg_not_stronglyreg13 = Graph("R~~~~~~~~~~~~~~~~~~~~~~~~~~~~w")
8447
distreg_not_stronglyreg13.name(new = "distreg_not_stronglyreg13")
8448
distreg_not_stronglyreg14 = Graph("O~~~~~~~~~~~~~~~~~~~~")
8449
distreg_not_stronglyreg14.name(new = "distreg_not_stronglyreg14")
8450
distreg_not_stronglyreg15 = Graph("K~~~~~~~~~~~")
8451
distreg_not_stronglyreg15.name(new = "distreg_not_stronglyreg15")
8452
distreg_not_stronglyreg16 = Graph("S~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{")
8453
distreg_not_stronglyreg16.name(new = "distreg_not_stronglyreg16")
8454
graph_objects.extend([distreg_not_stronglyreg7, distreg_not_stronglyreg8, distreg_not_stronglyreg9, distreg_not_stronglyreg10, distreg_not_stronglyreg11,
8455
distreg_not_stronglyreg12, distreg_not_stronglyreg13, distreg_not_stronglyreg14, distreg_not_stronglyreg15, distreg_not_stronglyreg16])
8456
all_graphs.extend([distreg_not_stronglyreg7, distreg_not_stronglyreg8, distreg_not_stronglyreg9, distreg_not_stronglyreg10, distreg_not_stronglyreg11,
8457
distreg_not_stronglyreg12, distreg_not_stronglyreg13, distreg_not_stronglyreg14, distreg_not_stronglyreg15, distreg_not_stronglyreg16])
8458
8459
###### DISCONNECTED GRAPHS ######
8460
8461
# Felix Lazebnik and Spencer Tofts, An Extremal Property of Turan Graphs. The electronic journal of combinatorics, 2010.
8462
LozTof10 = Graph('P????????????@?@?Y?@UBEO')
8463
LozTof10.name(new = "LozTof10")
8464
add_to_lists(LozTof10, non_connected_graphs)
8465
8466
# Product Graphs, fig. 5.4
8467
p5txp3 = graphs.PathGraph(5).tensor_product(graphs.PathGraph(3))
8468
p5txp3.name(new = "p5txp3")
8469
add_to_lists(p5txp3, non_connected_graphs)
8470
8471
add_to_lists(graphs.WorldMap(), non_connected_graphs)
8472
8473
8474
#graph_objects: all graphs with no duplicates
8475
#obvious way to remove duplicates in list of ALL objects
8476
8477
def remove_duplicates(seq, idfun=None):
8478
"""
8479
Utility method to remove duplicates from a list. Equality is determined
8480
based on the return value of the idfun parameter. If idfun is not given,
8481
the identity is used.
8482
8483
This method is taken from http://www.peterbe.com/plog/uniqifiers-benchmark
8484
8485
sage: graphs = [graphs.CompleteGraph(4), graphs.CompleteGraph(3), graphs.CycleGraph(4), graphs.CycleGraph(3)]
8486
sage: remove_duplicates(graphs, idfun=lambda g: g.canonical_label(algorithm='sage').graph6_string())
8487
Cycle graph was already in the list
8488
[Complete graph: Graph on 4 vertices,
8489
Complete graph: Graph on 3 vertices,
8490
Cycle graph: Graph on 4 vertices]
8491
"""
8492
if idfun is None:
8493
def idfun(x): return x
8494
seen = {}
8495
result = []
8496
for item in seq:
8497
marker = idfun(item)
8498
if marker in seen:
8499
print "{} was already in the list".format(item)
8500
continue
8501
seen[marker] = 1
8502
result.append(item)
8503
return result
8504
8505
def remove_duplicates_colours(seq, idfuns, verbose=False):
8506
"""
8507
Utility method to remove duplicates from a list. Equality is determined
8508
based on the return values of the idfuns parameter. This parameter expects
8509
to receive a list of functions (colours) which can be used to distinguish
8510
graphs. The only requirements are that calling any of the colours on isomorphic
8511
graphs should always return the same value, and that the last colour in
8512
the list should be different for non-isomorphic graphs.
8513
8514
This method is based on a method from http://www.peterbe.com/plog/uniqifiers-benchmark
8515
8516
sage: graphs = [graphs.CompleteGraph(4), graphs.CompleteGraph(3), graphs.CycleGraph(4), graphs.CycleGraph(3)]
8517
sage: remove_duplicates_colours(graphs, [Graph.order, (lambda g: max(g.degree())), lambda g: g.canonical_label(algorithm='sage').graph6_string()])
8518
[Complete graph: Graph on 4 vertices,
8519
Complete graph: Graph on 3 vertices,
8520
Cycle graph: Graph on 4 vertices]
8521
"""
8522
seen = {}
8523
result = []
8524
for item in seq:
8525
current = seen
8526
for pos, idfun in enumerate(idfuns):
8527
colour = idfun(item)
8528
if colour in current:
8529
if type(current[colour]) == dict:
8530
current = current[colour]
8531
elif pos == len(idfuns) - 1:
8532
if verbose: print "{} was already in the list. Previous version was {}.".format(item, current[colour])
8533
else:
8534
prev_item = current[colour]
8535
remaining_idfuns = idfuns[pos+1:]
8536
for pos2, idfun in enumerate(remaining_idfuns):
8537
colour1, colour2 = idfun(prev_item), idfun(item)
8538
if colour1 == colour2:
8539
if pos2 == len(remaining_idfuns) - 1:
8540
current[colour] = {colour1: prev_item}
8541
if verbose: print "{} was already in the list. Previous version was {}.".format(item, prev_item)
8542
else:
8543
current[colour] = {}
8544
current, colour = current[colour], colour1
8545
else:
8546
current[colour] = {colour1: prev_item, colour2: item}
8547
result.append(item)
8548
break
8549
break
8550
else:
8551
current[colour] = item
8552
result.append(item)
8553
break
8554
return result
8555
8556
8557
#could run this occasionally to check there are no duplicates
8558
#graph_objects = remove_duplicates(union_objects, idfun=g.canonical_label(algorithm='sage').graph6_string())
8559
# Class 0 pebbling graphs
8560
8561
class0graphs_dict = {
8562
'C05': 'IheA@GUAo',
8563
'C05C05': 'XheAHCPBGG?P?P?G_BG?O?@C?AG?AG?@e??OO?AH??Ga??PA??X',
8564
'C05C05C05': '~?@|heAHCPBGG?P?P?G_BG?O?@C?AG?AG?@e??OO?AH??Ga??PA??X_???G???H???@C???CG??@GG??A?C??@C@???P?G??AG?_??K_@???O?@???P??_??G_?G??AG?@???X??C??@???G??AG??G??AG??C??@C??@???X???GO?A????`??G_??@A??P???@A??P????`??K_???G???????@???@????C???C????G???G????G??@G????C??@?????@???P?????G??AG?????_??G_????@???X?????@???O??????_??G_?????G??AG?????@???P??????C??@c??????G??A???????G??AG??????C??@C??????@???P???????G??BG???????`??G???????@A??P???????@A??P????????`??G_???????GO?BG???????@????????????C???C????????G???G????????G???G????????C???c????????@???O?????????G??AG?????????_??G_????????@???P?????????@???X??????????_??G??????????G??AG?????????@???P??????????C??@C??????????G??BG??????????G??A???????????C??@C??????????@???P???????????G??AG???????????_??K_??????????@A??O???????????@A??P????????????`??G_???????????GO?AG???????????@A??X_???????????C???@????????????G???H????????????G???G_???????????C???CG???????????@???H@????????????G??A?C????????????_??G_G???????????@???P?G???????????@???P?C????????????_??K_@????????????G??A??G???????????@???P??_???????????C??@C?@????????????G??AG?@????????????G??BG??_???????????C??@???G???????????@???P??@????????????G??AG??C????????????_??G_??G???????????@???X???G???????????@A??O???C????????????`??G_??@????????????GO?AG???G???????????@A??P????_???????????CG?@c',
8565
'C05C05K2': 'qrh\\QmiT[?G@GAC@`@OGJ?ag@DW@TO?im?G?O@?H?C@A?GBA?GI@?CD_O@DOA?Gj?G?ig?O@T[?O@?A?G?_C_A?GAC?O@?WO@?CD?_A?GJ?_A?Gi?O@?CT_C?O@TO?_A?Ij_A?G?_@?C?O@?H?C?O@?O_A?G?_KG?_A?GI@?C?O@@WC?O@?CT?G?_A?Gj?G?_A?Ii?C?O@?DT_',
8566
'C05C05K3': '~?@J{S{aSfcaQcfccQQcc{??O?C_?W_?_O?SC?F?_C_A?Q_C?f?CCc?AAQ_?_cw?Ccc??QQS??cc{??_?A??O?C_?C?BC??_?_O?A?A__?C?F?_?C?c?O?A?Q_C??_Cw?_?CCc?A??OQS?C??_cw?C??cc_?A??QQS??_?Ccf_?C??_?A??O?A??c??_?C?BC??_?C?CA??O?A?A__?C??_?wC??_?C?c?O?A??OAS?_?C??_Cw?_?C??_c_?O?A??OQS?C??_?CCf??_?C??cc_?A??O?AQQ_?C??_?Ccf_?C??_?C??O?A??O?A??c??_?C??_?W_?C??_?C?CA??O?A??O?SC??_?C??_?wC??_?C??_C_A??O?A??OAS?_?C??_?C?f?C??_?C??_c_?O?A??O?AAQ_?_?C??_?CCf??_?C??_?Ccc??O?A??O?AQQ_?C??_?C??ccw',
8567
'C05C05K4': '~?@c~`HW}GPHDaNaGPCcPWaNaG`CPHCPWaG}???O??H??@a??FA??G@??C_O?@WA??N?G?G_?O?PG?O?PW?G?G{?A?aG??OCPG?@?PD_?A?aN??AGaG??@CPC_??PCPW??AGaN_??G??@???O??H???O??W_??G??[G??A??G@???O?@GC??@??D_G??A??N?G??A?AG?C??@?@C_@???O?PW?G??A?AN??_??GAG_?@???OCPG?@???OCPW??_??GAG{??G??AGaG??@???PCPG??C??@CPD_??G??AGaN_??G??A???O??C??@???c??@???O??W_??G??A??FA???_??G??_C??@???O?@GC??@???O?@WA???_??G??{?_??G??A?AG?C??@???O?PG?O??C??@?@D_?_??G??A?AN??_??G??A?aG??O??C??@?PC_?C??@???OCPW??_??G??A?aN??A???_??GaG_??C??@???PCPG??C??@???PCPW??A???_??GaG}???_??G??A???O??C??@???O??H???O??C??@??@a???_??G??A??FA???_??G??A??G@???O??C??@??C_O??C??@???O?@WA???_??G??A??N?G??A???_??G?G_?O??C??@???O?PG?O??C??@???O?PW?G??A???_??G?G{?A???_??G??A?aG??O??C??@???OCPG?@???O??C??@?PD_?A???_??G??A?aN??A???_??G??AGaG??@???O??C??@CPC_??O??C??@???PCPW??A???_??G??AGaN',
8568
'C05C05Q2': '~?@cr`HOmGPHDAJaGPCcPOaJaG`CPHCPOaGm???O??H??@A??BA??G@??C_O?@OA??J?G?G_?O?PG?O?PO?G?Gk?A?aG??OCPG?@?PD??A?aJ??AGaG??@CPC_??PCPO??AGaJ_??G??@???O??H???O??O_??G??KG??A??G@???O?@GC??@??D?G??A??J?G??A?AG?C??@?@C_@???O?PO?G??A?AJ??_??GAG_?@???OCPG?@???OCPO??_??GAGk??G??AGaG??@???PCPG??C??@CPD???G??AGaJ_??G??A???O??C??@???c??@???O??O_??G??A??BA???_??G??_C??@???O?@GC??@???O?@OA???_??G??k?_??G??A?AG?C??@???O?PG?O??C??@?@D??_??G??A?AJ??_??G??A?aG??O??C??@?PC_?C??@???OCPO??_??G??A?aJ??A???_??GaG_??C??@???PCPG??C??@???PCPO??A???_??GaGm???_??G??A???O??C??@???O??H???O??C??@??@A???_??G??A??BA???_??G??A??G@???O??C??@??C_O??C??@???O?@OA???_??G??A??J?G??A???_??G?G_?O??C??@???O?PG?O??C??@???O?PO?G??A???_??G?Gk?A???_??G??A?aG??O??C??@???OCPG?@???O??C??@?PD??A???_??G??A?aJ??A???_??G??AGaG??@???O??C??@CPC_??O??C??@???PCPO??A???_??G??AGaJ',
8569
'C05K02': 'IrGWOMAOW',
8570
'C05K03': 'N{Sw_SF?_A_F_COAc?w',
8571
'C05K04': 'S~`HW{G@GD_N?G?C_@W?N_?`?@H?@W_?{',
8572
'C05K05': 'X~}AHKVBwG?P?R?J_Bw?O?@C?AW?Aw?@}??OO?AH??Ha??VA??^',
8573
'C05K06': ']~~{ACbCwV_~?_?O_CW?f?A{?Fw?C??AC??b??Cw??V_??~_??_O??Oc??CW_??fA??A{C??Fw',
8574
'C05K07': 'b~~~{@@GWb`N@^?~_G?@@?CK?Gw?Hw?D{?@~??G???__?@B??@F???f_??Jw??@~_??C@???GH???GW_??C[G??@N@???JwC???~_',
8575
'C05K08': 'g~~~~}?OH@aFAN@N_VwB~?G??OG?OW?G[?AN??Rw?@^_?B~??A???@?_??OW??AF???G{???Rw???Vw???N}???A?O???OH???@@a???AFA???AN@???@N_O???VwA???B~',
8576
'C05K09': 'l~~~~~~_A?cBCFAF_bwC~?V{?~w?_??OC?CB??_w?AF_?C^??C~??A~_??~w??C????OC???_W???_w???O{???C^????fw???A~_???F~_???C?O???A?c????_W_???CFA????O{C????bwC????fwA????V{?_???F~',
8577
'C05K10': 'q~~~~~~~{?G@GBCB`@wG^?b{@Nw@^w?~{?G??@?G?C@_?GF??GN??CN_?@Fw??H~???n{??@~w??@?????_C???GB???@?w???CF_???G^????G~????C~_???@^w????N~_????_@????@?H????@?W_????_[G????GN@????@BwC????C^_G????H~?G????J~?C????F~_',
8578
'C05K11': 'v~~~~~~~~~o?O@G@a?wON@@{AFwANw@N{?V~?B~w?G???O@??OB??GB_?A@w??O^??@B{??ANw??A^w??@^{???^~???A?????G?_???OB????OF????GF_???ABw????O~????@F{????A^w????A~w????@~}?????O?O????A?H?????G@a?????OFA?????ON@?????GN_O????AFwA?????P~?G????@N{?O????A~w?O????B~w',
8579
'C05K12': '{~~~~~~~~~~~_?O?c?W_FA?{CBwCFwAF{?b~?C~w?V~_?~~??_???O?_?C?W??_F??A?{??CBw??CFw??AF{???b~???C~w???V~_???~~????_?????O?_???C?W????_F????A?{????CBw????CFw????AF{?????b~?????C~w?????V~_?????~~_?????_?O?????O?c?????C?W_?????_FA?????A?{C?????CBwC?????CFwA?????AF{?_?????b~?C?????C~w?O?????V~_?_?????~~',
8580
'C05K13': '~?@@~~~~~~~~~~~~~_?G?H?BC?[G@wGBwCB{@@~?G^w?b~_@N~?@^~??~~_?G???@?@??C?K??G?w??G@w??C@{??@?~???GNw???`~_??@F~???@N~????n~_???N~w???@??????C?C????G?W????G?w????C?{????@?^?????GFw?????_~_????@B~?????@F~??????f~_?????J~w?????@~~_?????C?@??????G?H??????G?W_?????C?[G?????@?N@??????GBwC??????_^_G?????@@~?G?????@B~?C??????b~_@??????H~w?G?????@^~??_?????F~{',
8581
'C05K14': '~?@E~~~~~~~~~~~~~~~o?A?@G?KO?wO@wG@{A?~?ONw@@~_AF~?AN~?@N~_?V~w?B~~??G????O?G??O?W??G?[??A?N???OBw??@?^_??A@~???AB~???@B~_???P~w???A^~????J~{????^~w????O??????G?C????A?B?????O?w????@?F_????A?^?????A?~?????@?~_?????O^w?????AF~??????G~{??????R~w??????V~w??????N~}??????A??O??????O?H??????@?@a??????A?FA??????A?N@??????@?N_O??????OFwA??????A@~?G??????GN{?O??????O~w?O??????P~w?G??????H~{?A??????A~~??O??????^~w',
8582
'C05K15': '~?@J~~~~~~~~~~~~~~~~~{??O?C_?W_?wO?{C?^?_FwA?~_CB~?CF~?AF~_?b~w?C~~??V~{??~~w??_????O?C??C?B???_?w??A?F_??C?^???C?~???A?~_???_^w???CF~????O~{????b~w????f~w????V~{????F~~?????_??????A??_????C?B?????C?F?????A?F_?????_Bw?????C?~??????OF{??????_^w??????_~w??????O~{??????C^~???????f~w??????A~~_??????F~~_??????C??O??????A??c???????_?W_??????C?FA???????O?{C???????_BwC???????_FwA???????OF{?_??????CB~?C???????_~w?O??????AF~_?_??????C^~??_??????C~~??O??????A~~_?C???????~~w',
8583
'C05K16': '~?@O~~~~~~~~~~~~~~~~~~~~_?@??H??W_?[G?N@?BwC?^_G@~?GB~?CB~_@@~w?G^~??b~{?@N~w?@^~w??~~{??G????@??G??C?@_??G?F???G?N???C?N_??@?Fw???G@~????_N{???@?~w???@@~w????`~{????G~~????@N~w????D~~_????N~~?????G???????C??_????@??W?????G?F??????_?{?????@?Bw?????@?Fw??????_F{??????GB~??????@?~w??????CF~_??????G^~???????G~~???????C~~_??????@^~w???????N~~_???????_?@???????@??H???????@??W_???????_?[G???????G?N@???????@?BwC???????C?^_G???????G@~?G???????GB~?C???????CB~_@???????@@~w?G???????G^~??_???????b~{?@???????@N~w?@???????@^~w??_???????~~{',
8584
'C05K17': '~?@T~~~~~~~~~~~~~~~~~~~~~~}??A??H??KO?FA?@wG?N_O?~?O@~?G@~_A?~w?ON~?@@~{?AF~w?AN~w?@N~{??V~~??B~~w??G?????O?@???O?B???G?B_??A?@w???O?^???@?B{???A?Nw???A?^w???@?^{????ON~????AB~w????G^~_????P~~?????R~~?????J~~_????B~~w?????O???????@??C?????A??W?????A??w?????@??{??????O?^??????A?Fw??????G?~_??????OB~???????OF~???????GF~_??????AB~w???????O~~???????@F~{???????A^~w???????A~~w???????@~~}????????O??O???????A??H????????G?@a????????O?FA????????O?N@????????G?N_O???????A?FwA????????O@~?G???????@?N{?O???????A?~w?O???????A@~w?G???????@@~{?A????????O~~??O???????AN~w?@????????H~~_?A????????V~~??A????????^~~',
8585
'C05K18': '~?@Y~~~~~~~~~~~~~~~~~~~~~~~~~{??A??C_?BC??wO?F__?^?_?~?O?~_C?^w?_F~?A?~{?CB~w?CF~w?AF~{??b~~??C~~w??V~~_??~~~???_?????O??_??C??W???_?F???A??{???C?Bw???C?Fw???A?F{????_B~????C?~w????OF~_????_^~?????_~~?????O~~_????C^~w?????f~~?????A~~{?????F~~w?????C????????A??C??????_?B??????C??w??????O?F_??????_?^???????_?~???????O?~_??????C?^w???????_F~???????A?~{???????CB~w???????CF~w???????AF~{????????b~~????????C~~w????????V~~_????????~~~_????????_??O????????O??c????????C??W_????????_?FA????????A??{C????????C?BwC????????C?FwA????????A?F{?_????????_B~?C????????C?~w?O????????OF~_?_????????_^~??_????????_~~??O????????O~~_?C????????C^~w??_????????f~~??A????????A~~{??C????????F~~w',
8586
'C05K19': '~?@^~~~~~~~~~~~~~~~~~~~~~~~~~~~~{??@??@G??W_?B`??N@??^?_?^_G?Nw@?B~?C?^{?G@~w?GB~w?CB~{?@@~~??G^~w??b~~_?@N~~??@^~~???~~~_??G?????@??@???C??K???G??w???G?@w???C?@{???@??~????G?Nw????_@~_???@?F~????@?N~?????_N~_????GF~w????@@~~?????CN~{?????G~~w?????H~~w?????D~~{?????@~~~??????G?????????_??_?????@??B??????@??F???????_?F_??????G?Bw??????@??~???????C?F{???????G?^w???????G?~w???????C?~{???????@?^~????????GF~w????????_~~_???????@B~~????????@F~~?????????f~~_????????J~~w????????@~~~_????????C??@?????????G??H?????????G??W_????????C??[G????????@??N@?????????G?BwC?????????_?^_G????????@?@~?G????????@?B~?C?????????_B~_@?????????G@~w?G????????@?^~??_????????CB~{?@?????????GN~w?@?????????G^~w??_????????C^~{??G????????@N~~??@?????????J~~w??C?????????~~~_',
8587
'C05K20': '~?@c~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}???O??H??@a??FA??N@??N_O?FwA?@~?G?N{?O?~w?O@~w?G@~{?A?~~??ON~w?@@~~_?AF~~??AN~~??@N~~_??V~~w??B~~~???G??????O??G???O??W???G??[???A??N????O?Bw???@??^_???A?@~????A?B~????@?B~_????O@~w????A?^~?????GB~{?????ON~w?????O^~w?????G^~{?????AN~~??????R~~w?????@^~~_?????B~~~??????A?????????@???_??????O??W??????A??F???????G??{???????O?Bw???????O?Fw???????G?F{???????A?B~????????O?~w???????@?F~_???????A?^~????????A?~~????????@?~~_????????O^~w????????AF~~?????????G~~{?????????R~~w?????????V~~w?????????N~~}?????????A???O?????????O??H?????????@??@a?????????A??FA?????????A??N@?????????@??N_O?????????O?FwA?????????A?@~?G?????????G?N{?O?????????O?~w?O?????????O@~w?G?????????G@~{?A?????????A?~~??O?????????ON~w?@?????????@@~~_?A?????????AF~~??A?????????AN~~??@?????????@N~~_??O?????????V~~w??A?????????B~~~',
8588
'C05Q2': 'Sr`HOkG@GD?J?G?C_@O?J_?`?@H?@O_?k',
8589
'C05Q3': 'gr`HOm?OH@ABAG@C_POAJ?G??OG?OO?GK?AG??PG?@D??AJ??A???@?_??OO??AB???G_???PG???PO???Gm???A?O???OH???@@A???ABA???AG@???@C_O???POA???AJ',
8590
'C05Q4': '~?@Or`HOm?OH@ABAG@C_POAJ_?@??H??O_?KG?G@?@GC?D?G?J?GA??C@?_@?OO?GAB??_G_?@?PG?@?PO??_Gk??G????@??G??C?@???G?B???G?G???C?C_??@?@O???G?J????_G????@?OG???@?OO????_GK????GAG????@?PG????C@D?????GAJ?????G???????C??_????@??O?????G?B??????_?_?????@?@G?????@?@O??????_?k??????GA???????@?OG??????C@@???????GAB???????GAG???????C@C_??????@?PO???????GAJ_???????_?@???????@??H???????@??O_???????_?KG???????G?G@???????@?@GC???????C?D?G???????G?J?G???????GA??C???????C@?_@???????@?OO?G???????GAB??_???????_G_?@???????@?PG?@???????@?PO??_???????_Gk',
8591
'C06Pow02': 'EznW',
8592
'C07Pow02': 'FzM]W',
8593
'C08Pow02': 'GzK[]K',
8594
'C09Pow02': 'HzKW[NB',
8595
'C10Pow04': 'I~||}~n|w',
8596
'C11Pow04': 'J~|x{~Nx~f_',
8597
'C12Pow04': 'K~|xw}Np~F}N',
8598
'C12Pow05': 'K~~z|~^z~n~^',
8599
'C13Pow04': 'L~|xw{N`}F{N{N',
8600
'C14Pow04': 'M~|xw{N@{FwNwN{F_',
8601
'C15Pow04': 'N~|xw{N@wFoNoNwF}@w',
8602
'C16Pow04': 'O~|xw{N@wF_N_NoF{@~_N',
8603
'C17Pow04': 'P~|xw{N@wF_N?N_Fw@~?N{?{',
8604
'C18Pow04': 'Q~|xw{N@wF_N?N?Fo@}?Nw?~o@w',
8605
'C19Pow04': 'R~|xw{N@wF_N?N?F_@{?No?~_@~_@w',
8606
'C20Pow04': 'S~|xw{N@wF_N?N?F_@w?N_?~?@~?@~_?{',
8607
'C21Pow04': 'T~|xw{N@wF_N?N?F_@w?N??}?@}?@~??~o?N',
8608
'C22Pow04': 'U~|xw{N@wF_N?N?F_@w?N??{?@{?@}??~_?N{?@w',
8609
'C23Pow04': 'V~|xw{N@wF_N?N?F_@w?N??{?@w?@{??~??Nw?@~_?F_',
8610
'C24Pow08': 'W~~~~~~^x~b~B~@~_^wB~_N~?^~?^~_N~wB~~?^~{@~~wB~',
8611
'C25Pow08': 'X~~~~~~^x~b~B~@~_^wB~?N}?^}?^~?N~oB~}?^~w@~~oB~~oB~',
8612
'C26Pow08': 'Y~~~~~~^x~b~B~@~_^wB~?N{?^{?^}?N~_B~{?^~o@~~_B~~_B~~o@~_',
8613
'C27Pow08': 'Z~~~~~~^x~b~B~@~_^wB~?N{?^w?^{?N~?B~w?^~_@~~?B~~?B~~_@~~w?^w',
8614
'C28Pow08': '[~~~~~~^x~b~B~@~_^wB~?N{?^w?^w?N}?B~o?^~?@~}?B~}?B~~?@~~o?^~}?B~',
8615
'C29Pow08': '\\~~~~~~^x~b~B~@~_^wB~?N{?^w?^w?N{?B~_?^}?@~{?B~{?B~}?@~~_?^~{?B~~o?N{',
8616
'C30Pow08': ']~~~~~~^x~b~B~@~_^wB~?N{?^w?^w?N{?B~??^{?@~w?B~w?B~{?@~~??^~w?B~~_?N~~??^w',
8617
'C31Pow08': '^~~~~~~^x~b~B~@~_^wB~?N{?^w?^w?N{?B~??^w?@~o?B~o?B~w?@~}??^~o?B~~??N~}??^~}??^w',
8618
'C32Pow08': '_~~~~~~^x~b~B~@~_^wB~?N{?^w?^w?N{?B~??^w?@~_?B~_?B~o?@~{??^~_?B~}??N~{??^~{??^~}??N{',
8619
'C33Pow08': '`~~~~~~^x~b~B~@~_^wB~?N{?^w?^w?N{?B~??^w?@~_?B~??B~_?@~w??^~??B~{??N~w??^~w??^~{??N~~??B~',
8620
'C34Pow08': 'a~~~~~~^x~b~B~@~_^wB~?N{?^w?^w?N{?B~??^w?@~_?B~??B~??@~o??^}??B~w??N~o??^~o??^~w??N~}??B~~o??^w',
8621
'C35Pow08': 'b~~~~~~^x~b~B~@~_^wB~?N{?^w?^w?N{?B~??^w?@~_?B~??B~??@~_??^{??B~o??N~_??^~_??^~o??N~{??B~~_??^~}??@~_',
8622
'C36Pow08': 'c~~~~~~^x~b~B~@~_^wB~?N{?^w?^w?N{?B~??^w?@~_?B~??B~??@~_??^w??B~_??N~???^~???^~_??N~w??B~~???^~{??@~~w??B~',
8623
'C37Pow08': 'd~~~~~~^x~b~B~@~_^wB~?N{?^w?^w?N{?B~??^w?@~_?B~??B~??@~_??^w??B~???N}???^}???^~???N~o??B~}???^~w??@~~o??B~~o??B~',
8624
'C38Pow08': 'e~~~~~~^x~b~B~@~_^wB~?N{?^w?^w?N{?B~??^w?@~_?B~??B~??@~_??^w??B~???N{???^{???^}???N~_??B~{???^~o??@~~_??B~~_??B~~o??@~_',
8625
'C39Pow08': 'f~~~~~~^x~b~B~@~_^wB~?N{?^w?^w?N{?B~??^w?@~_?B~??B~??@~_??^w??B~???N{???^w???^{???N~???B~w???^~_??@~~???B~~???B~~_??@~~w???^w',
8626
'C40Pow08': 'g~~~~~~^x~b~B~@~_^wB~?N{?^w?^w?N{?B~??^w?@~_?B~??B~??@~_??^w??B~???N{???^w???^w???N}???B~o???^~???@~}???B~}???B~~???@~~o???^~}???B~',
8627
'C41Pow08': 'h~~~~~~^x~b~B~@~_^wB~?N{?^w?^w?N{?B~??^w?@~_?B~??B~??@~_??^w??B~???N{???^w???^w???N{???B~_???^}???@~{???B~{???B~}???@~~_???^~{???B~~o???N{',
8628
'C42Pow08': 'i~~~~~~^x~b~B~@~_^wB~?N{?^w?^w?N{?B~??^w?@~_?B~??B~??@~_??^w??B~???N{???^w???^w???N{???B~????^{???@~w???B~w???B~{???@~~????^~w???B~~_???N~~????^w',
8629
'C43Pow08': 'j~~~~~~^x~b~B~@~_^wB~?N{?^w?^w?N{?B~??^w?@~_?B~??B~??@~_??^w??B~???N{???^w???^w???N{???B~????^w???@~o???B~o???B~w???@~}????^~o???B~~????N~}????^~}????^w',
8630
'C44Pow08': 'k~~~~~~^x~b~B~@~_^wB~?N{?^w?^w?N{?B~??^w?@~_?B~??B~??@~_??^w??B~???N{???^w???^w???N{???B~????^w???@~_???B~_???B~o???@~{????^~_???B~}????N~{????^~{????^~}????N{',
8631
'C45Pow08': 'l~~~~~~^x~b~B~@~_^wB~?N{?^w?^w?N{?B~??^w?@~_?B~??B~??@~_??^w??B~???N{???^w???^w???N{???B~????^w???@~_???B~????B~_???@~w????^~????B~{????N~w????^~w????^~{????N~~????B~',
8632
'C46Pow08': 'm~~~~~~^x~b~B~@~_^wB~?N{?^w?^w?N{?B~??^w?@~_?B~??B~??@~_??^w??B~???N{???^w???^w???N{???B~????^w???@~_???B~????B~????@~o????^}????B~w????N~o????^~o????^~w????N~}????B~~o????^w',
8633
'C47Pow08': 'n~~~~~~^x~b~B~@~_^wB~?N{?^w?^w?N{?B~??^w?@~_?B~??B~??@~_??^w??B~???N{???^w???^w???N{???B~????^w???@~_???B~????B~????@~_????^{????B~o????N~_????^~_????^~o????N~{????B~~_????^~}????@~_',
8634
'C48Pow08': 'o~~~~~~^x~b~B~@~_^wB~?N{?^w?^w?N{?B~??^w?@~_?B~??B~??@~_??^w??B~???N{???^w???^w???N{???B~????^w???@~_???B~????B~????@~_????^w????B~_????N~?????^~?????^~_????N~w????B~~?????^~{????@~~w????B~',
8635
'C49Pow08': 'p~~~~~~^x~b~B~@~_^wB~?N{?^w?^w?N{?B~??^w?@~_?B~??B~??@~_??^w??B~???N{???^w???^w???N{???B~????^w???@~_???B~????B~????@~_????^w????B~?????N}?????^}?????^~?????N~o????B~}?????^~w????@~~o????B~~o????B~',
8636
'C50Pow08': 'q~~~~~~^x~b~B~@~_^wB~?N{?^w?^w?N{?B~??^w?@~_?B~??B~??@~_??^w??B~???N{???^w???^w???N{???B~????^w???@~_???B~????B~????@~_????^w????B~?????N{?????^{?????^}?????N~_????B~{?????^~o????@~~_????B~~_????B~~o????@~_',
8637
'C51Pow08': 'r~~~~~~^x~b~B~@~_^wB~?N{?^w?^w?N{?B~??^w?@~_?B~??B~??@~_??^w??B~???N{???^w???^w???N{???B~????^w???@~_???B~????B~????@~_????^w????B~?????N{?????^w?????^{?????N~?????B~w?????^~_????@~~?????B~~?????B~~_????@~~w?????^w',
8638
'C52Pow08': 's~~~~~~^x~b~B~@~_^wB~?N{?^w?^w?N{?B~??^w?@~_?B~??B~??@~_??^w??B~???N{???^w???^w???N{???B~????^w???@~_???B~????B~????@~_????^w????B~?????N{?????^w?????^w?????N}?????B~o?????^~?????@~}?????B~}?????B~~?????@~~o?????^~}?????B~',
8639
'C53Pow08': 't~~~~~~^x~b~B~@~_^wB~?N{?^w?^w?N{?B~??^w?@~_?B~??B~??@~_??^w??B~???N{???^w???^w???N{???B~????^w???@~_???B~????B~????@~_????^w????B~?????N{?????^w?????^w?????N{?????B~_?????^}?????@~{?????B~{?????B~}?????@~~_?????^~{?????B~~o?????N{',
8640
'C54Pow08': 'u~~~~~~^x~b~B~@~_^wB~?N{?^w?^w?N{?B~??^w?@~_?B~??B~??@~_??^w??B~???N{???^w???^w???N{???B~????^w???@~_???B~????B~????@~_????^w????B~?????N{?????^w?????^w?????N{?????B~??????^{?????@~w?????B~w?????B~{?????@~~??????^~w?????B~~_?????N~~??????^w',
8641
'C55Pow08': 'v~~~~~~^x~b~B~@~_^wB~?N{?^w?^w?N{?B~??^w?@~_?B~??B~??@~_??^w??B~???N{???^w???^w???N{???B~????^w???@~_???B~????B~????@~_????^w????B~?????N{?????^w?????^w?????N{?????B~??????^w?????@~o?????B~o?????B~w?????@~}??????^~o?????B~~??????N~}??????^~}??????^w',
8642
'C56Pow08': 'w~~~~~~^x~b~B~@~_^wB~?N{?^w?^w?N{?B~??^w?@~_?B~??B~??@~_??^w??B~???N{???^w???^w???N{???B~????^w???@~_???B~????B~????@~_????^w????B~?????N{?????^w?????^w?????N{?????B~??????^w?????@~_?????B~_?????B~o?????@~{??????^~_?????B~}??????N~{??????^~{??????^~}??????N{',
8643
'C57Pow08': 'x~~~~~~^x~b~B~@~_^wB~?N{?^w?^w?N{?B~??^w?@~_?B~??B~??@~_??^w??B~???N{???^w???^w???N{???B~????^w???@~_???B~????B~????@~_????^w????B~?????N{?????^w?????^w?????N{?????B~??????^w?????@~_?????B~??????B~_?????@~w??????^~??????B~{??????N~w??????^~w??????^~{??????N~~??????B~',
8644
'C58Pow08': 'y~~~~~~^x~b~B~@~_^wB~?N{?^w?^w?N{?B~??^w?@~_?B~??B~??@~_??^w??B~???N{???^w???^w???N{???B~????^w???@~_???B~????B~????@~_????^w????B~?????N{?????^w?????^w?????N{?????B~??????^w?????@~_?????B~??????B~??????@~o??????^}??????B~w??????N~o??????^~o??????^~w??????N~}??????B~~o??????^w',
8645
'C59Pow08': 'z~~~~~~^x~b~B~@~_^wB~?N{?^w?^w?N{?B~??^w?@~_?B~??B~??@~_??^w??B~???N{???^w???^w???N{???B~????^w???@~_???B~????B~????@~_????^w????B~?????N{?????^w?????^w?????N{?????B~??????^w?????@~_?????B~??????B~??????@~_??????^{??????B~o??????N~_??????^~_??????^~o??????N~{??????B~~_??????^~}??????@~_',
8646
'C60Pow16': '{~~~~~~~~~~~~~~~~~~~~~~|~~x~~w~~{N~~@~~wF~~_N~~?N~~?F~~_@~~w?N~~??~~{?@~~w?@~~w??~~{??N~~??@~~w??F~~_??N~~???N~~???F~~_??@~~w???N~~????~~{???@~~w???@~~w????~~}????N~~o???@~~~????F~~}????N~~}????N~~~????F~~~o???@~~~}????N~~~w????~~~~o???@~~~~o???@~~~~w????~~~~}????N~~~~o???@~~~~~????F~~~~}????N~~',
8647
'C61Pow16': '|~~~~~~~~~~~~~~~~~~~~~~|~~x~~w~~{N~~@~~wF~~_N~~?N~~?F~~_@~~w?N~~??~~{?@~~w?@~~w??~~{??N~~??@~~w??F~~_??N~~???N~~???F~~_??@~~w???N~~????~~{???@~~w???@~~w????~~{????N~~_???@~~}????F~~{????N~~{????N~~}????F~~~_???@~~~{????N~~~o????~~~~_???@~~~~_???@~~~~o????~~~~{????N~~~~_???@~~~~}????F~~~~{????N~~~~{????N~~',
8648
'C62Pow16': '}~~~~~~~~~~~~~~~~~~~~~~|~~x~~w~~{N~~@~~wF~~_N~~?N~~?F~~_@~~w?N~~??~~{?@~~w?@~~w??~~{??N~~??@~~w??F~~_??N~~???N~~???F~~_??@~~w???N~~????~~{???@~~w???@~~w????~~{????N~~????@~~{????F~~w????N~~w????N~~{????F~~~????@~~~w????N~~~_????~~~~????@~~~~????@~~~~_????~~~~w????N~~~~????@~~~~{????F~~~~w????N~~~~w????N~~~~{????F~~_',
8649
'C63Pow16': '~??~~~~~~~~~~~~~~~~~~~~~~~|~~x~~w~~{N~~@~~wF~~_N~~?N~~?F~~_@~~w?N~~??~~{?@~~w?@~~w??~~{??N~~??@~~w??F~~_??N~~???N~~???F~~_??@~~w???N~~????~~{???@~~w???@~~w????~~{????N~~????@~~w????F~~o????N~~o????N~~w????F~~}????@~~~o????N~~~?????~~~}????@~~~}????@~~~~?????~~~~o????N~~~}????@~~~~w????F~~~~o????N~~~~o????N~~~~w????F~~~~}????@~~w',
8650
'C64Pow16': '~?@?~~~~~~~~~~~~~~~~~~~~~~|~~x~~w~~{N~~@~~wF~~_N~~?N~~?F~~_@~~w?N~~??~~{?@~~w?@~~w??~~{??N~~??@~~w??F~~_??N~~???N~~???F~~_??@~~w???N~~????~~{???@~~w???@~~w????~~{????N~~????@~~w????F~~_????N~~_????N~~o????F~~{????@~~~_????N~~}?????~~~{????@~~~{????@~~~}?????~~~~_????N~~~{????@~~~~o????F~~~~_????N~~~~_????N~~~~o????F~~~~{????@~~~~~_????N~~',
8651
'C65Pow16': '~?@@~~~~~~~~~~~~~~~~~~~~~~|~~x~~w~~{N~~@~~wF~~_N~~?N~~?F~~_@~~w?N~~??~~{?@~~w?@~~w??~~{??N~~??@~~w??F~~_??N~~???N~~???F~~_??@~~w???N~~????~~{???@~~w???@~~w????~~{????N~~????@~~w????F~~_????N~~?????N~~_????F~~w????@~~~?????N~~{?????~~~w????@~~~w????@~~~{?????~~~~?????N~~~w????@~~~~_????F~~~~?????N~~~~?????N~~~~_????F~~~~w????@~~~~~?????N~~~~{?????~~{',
8652
'C66Pow16': '~?@A~~~~~~~~~~~~~~~~~~~~~~|~~x~~w~~{N~~@~~wF~~_N~~?N~~?F~~_@~~w?N~~??~~{?@~~w?@~~w??~~{??N~~??@~~w??F~~_??N~~???N~~???F~~_??@~~w???N~~????~~{???@~~w???@~~w????~~{????N~~????@~~w????F~~_????N~~?????N~~?????F~~o????@~~}?????N~~w?????~~~o????@~~~o????@~~~w?????~~~}?????N~~~o????@~~~~?????F~~~}?????N~~~}?????N~~~~?????F~~~~o????@~~~~}?????N~~~~w?????~~~~~o????@~~w',
8653
'C67Pow16': '~?@B~~~~~~~~~~~~~~~~~~~~~~|~~x~~w~~{N~~@~~wF~~_N~~?N~~?F~~_@~~w?N~~??~~{?@~~w?@~~w??~~{??N~~??@~~w??F~~_??N~~???N~~???F~~_??@~~w???N~~????~~{???@~~w???@~~w????~~{????N~~????@~~w????F~~_????N~~?????N~~?????F~~_????@~~{?????N~~o?????~~~_????@~~~_????@~~~o?????~~~{?????N~~~_????@~~~}?????F~~~{?????N~~~{?????N~~~}?????F~~~~_????@~~~~{?????N~~~~o?????~~~~~_????@~~~~~_????@~~w',
8654
'C68Pow16': '~?@C~~~~~~~~~~~~~~~~~~~~~~|~~x~~w~~{N~~@~~wF~~_N~~?N~~?F~~_@~~w?N~~??~~{?@~~w?@~~w??~~{??N~~??@~~w??F~~_??N~~???N~~???F~~_??@~~w???N~~????~~{???@~~w???@~~w????~~{????N~~????@~~w????F~~_????N~~?????N~~?????F~~_????@~~w?????N~~_?????~~~?????@~~~?????@~~~_?????~~~w?????N~~~?????@~~~{?????F~~~w?????N~~~w?????N~~~{?????F~~~~?????@~~~~w?????N~~~~_?????~~~~~?????@~~~~~?????@~~~~~_?????~~{',
8655
'C69Pow16': '~?@D~~~~~~~~~~~~~~~~~~~~~~|~~x~~w~~{N~~@~~wF~~_N~~?N~~?F~~_@~~w?N~~??~~{?@~~w?@~~w??~~{??N~~??@~~w??F~~_??N~~???N~~???F~~_??@~~w???N~~????~~{???@~~w???@~~w????~~{????N~~????@~~w????F~~_????N~~?????N~~?????F~~_????@~~w?????N~~??????~~}?????@~~}?????@~~~??????~~~o?????N~~}?????@~~~w?????F~~~o?????N~~~o?????N~~~w?????F~~~}?????@~~~~o?????N~~~~??????~~~~}?????@~~~~}?????@~~~~~??????~~~~~o?????N~~',
8656
'C70Pow16': '~?@E~~~~~~~~~~~~~~~~~~~~~~|~~x~~w~~{N~~@~~wF~~_N~~?N~~?F~~_@~~w?N~~??~~{?@~~w?@~~w??~~{??N~~??@~~w??F~~_??N~~???N~~???F~~_??@~~w???N~~????~~{???@~~w???@~~w????~~{????N~~????@~~w????F~~_????N~~?????N~~?????F~~_????@~~w?????N~~??????~~{?????@~~{?????@~~}??????~~~_?????N~~{?????@~~~o?????F~~~_?????N~~~_?????N~~~o?????F~~~{?????@~~~~_?????N~~~}??????~~~~{?????@~~~~{?????@~~~~}??????~~~~~_?????N~~~~{?????@~~w',
8657
'C71Pow16': '~?@F~~~~~~~~~~~~~~~~~~~~~~|~~x~~w~~{N~~@~~wF~~_N~~?N~~?F~~_@~~w?N~~??~~{?@~~w?@~~w??~~{??N~~??@~~w??F~~_??N~~???N~~???F~~_??@~~w???N~~????~~{???@~~w???@~~w????~~{????N~~????@~~w????F~~_????N~~?????N~~?????F~~_????@~~w?????N~~??????~~{?????@~~w?????@~~{??????~~~??????N~~w?????@~~~_?????F~~~??????N~~~??????N~~~_?????F~~~w?????@~~~~??????N~~~{??????~~~~w?????@~~~~w?????@~~~~{??????~~~~~??????N~~~~w?????@~~~~~_?????F~~_',
8658
'C72Pow16': '~?@G~~~~~~~~~~~~~~~~~~~~~~|~~x~~w~~{N~~@~~wF~~_N~~?N~~?F~~_@~~w?N~~??~~{?@~~w?@~~w??~~{??N~~??@~~w??F~~_??N~~???N~~???F~~_??@~~w???N~~????~~{???@~~w???@~~w????~~{????N~~????@~~w????F~~_????N~~?????N~~?????F~~_????@~~w?????N~~??????~~{?????@~~w?????@~~w??????~~}??????N~~o?????@~~~??????F~~}??????N~~}??????N~~~??????F~~~o?????@~~~}??????N~~~w??????~~~~o?????@~~~~o?????@~~~~w??????~~~~}??????N~~~~o?????@~~~~~??????F~~~~}??????N~~',
8659
'C73Pow16': '~?@H~~~~~~~~~~~~~~~~~~~~~~|~~x~~w~~{N~~@~~wF~~_N~~?N~~?F~~_@~~w?N~~??~~{?@~~w?@~~w??~~{??N~~??@~~w??F~~_??N~~???N~~???F~~_??@~~w???N~~????~~{???@~~w???@~~w????~~{????N~~????@~~w????F~~_????N~~?????N~~?????F~~_????@~~w?????N~~??????~~{?????@~~w?????@~~w??????~~{??????N~~_?????@~~}??????F~~{??????N~~{??????N~~}??????F~~~_?????@~~~{??????N~~~o??????~~~~_?????@~~~~_?????@~~~~o??????~~~~{??????N~~~~_?????@~~~~}??????F~~~~{??????N~~~~{??????N~~',
8660
'C74Pow16': '~?@I~~~~~~~~~~~~~~~~~~~~~~|~~x~~w~~{N~~@~~wF~~_N~~?N~~?F~~_@~~w?N~~??~~{?@~~w?@~~w??~~{??N~~??@~~w??F~~_??N~~???N~~???F~~_??@~~w???N~~????~~{???@~~w???@~~w????~~{????N~~????@~~w????F~~_????N~~?????N~~?????F~~_????@~~w?????N~~??????~~{?????@~~w?????@~~w??????~~{??????N~~??????@~~{??????F~~w??????N~~w??????N~~{??????F~~~??????@~~~w??????N~~~_??????~~~~??????@~~~~??????@~~~~_??????~~~~w??????N~~~~??????@~~~~{??????F~~~~w??????N~~~~w??????N~~~~{??????F~~_',
8661
'C75Pow16': '~?@J~~~~~~~~~~~~~~~~~~~~~~|~~x~~w~~{N~~@~~wF~~_N~~?N~~?F~~_@~~w?N~~??~~{?@~~w?@~~w??~~{??N~~??@~~w??F~~_??N~~???N~~???F~~_??@~~w???N~~????~~{???@~~w???@~~w????~~{????N~~????@~~w????F~~_????N~~?????N~~?????F~~_????@~~w?????N~~??????~~{?????@~~w?????@~~w??????~~{??????N~~??????@~~w??????F~~o??????N~~o??????N~~w??????F~~}??????@~~~o??????N~~~???????~~~}??????@~~~}??????@~~~~???????~~~~o??????N~~~}??????@~~~~w??????F~~~~o??????N~~~~o??????N~~~~w??????F~~~~}??????@~~w',
8662
'C76Pow16': '~?@K~~~~~~~~~~~~~~~~~~~~~~|~~x~~w~~{N~~@~~wF~~_N~~?N~~?F~~_@~~w?N~~??~~{?@~~w?@~~w??~~{??N~~??@~~w??F~~_??N~~???N~~???F~~_??@~~w???N~~????~~{???@~~w???@~~w????~~{????N~~????@~~w????F~~_????N~~?????N~~?????F~~_????@~~w?????N~~??????~~{?????@~~w?????@~~w??????~~{??????N~~??????@~~w??????F~~_??????N~~_??????N~~o??????F~~{??????@~~~_??????N~~}???????~~~{??????@~~~{??????@~~~}???????~~~~_??????N~~~{??????@~~~~o??????F~~~~_??????N~~~~_??????N~~~~o??????F~~~~{??????@~~~~~_??????N~~',
8663
'C77Pow16': '~?@L~~~~~~~~~~~~~~~~~~~~~~|~~x~~w~~{N~~@~~wF~~_N~~?N~~?F~~_@~~w?N~~??~~{?@~~w?@~~w??~~{??N~~??@~~w??F~~_??N~~???N~~???F~~_??@~~w???N~~????~~{???@~~w???@~~w????~~{????N~~????@~~w????F~~_????N~~?????N~~?????F~~_????@~~w?????N~~??????~~{?????@~~w?????@~~w??????~~{??????N~~??????@~~w??????F~~_??????N~~???????N~~_??????F~~w??????@~~~???????N~~{???????~~~w??????@~~~w??????@~~~{???????~~~~???????N~~~w??????@~~~~_??????F~~~~???????N~~~~???????N~~~~_??????F~~~~w??????@~~~~~???????N~~~~{???????~~{',
8664
'C78Pow16': '~?@M~~~~~~~~~~~~~~~~~~~~~~|~~x~~w~~{N~~@~~wF~~_N~~?N~~?F~~_@~~w?N~~??~~{?@~~w?@~~w??~~{??N~~??@~~w??F~~_??N~~???N~~???F~~_??@~~w???N~~????~~{???@~~w???@~~w????~~{????N~~????@~~w????F~~_????N~~?????N~~?????F~~_????@~~w?????N~~??????~~{?????@~~w?????@~~w??????~~{??????N~~??????@~~w??????F~~_??????N~~???????N~~???????F~~o??????@~~}???????N~~w???????~~~o??????@~~~o??????@~~~w???????~~~}???????N~~~o??????@~~~~???????F~~~}???????N~~~}???????N~~~~???????F~~~~o??????@~~~~}???????N~~~~w???????~~~~~o??????@~~w',
8665
'C79Pow16': '~?@N~~~~~~~~~~~~~~~~~~~~~~|~~x~~w~~{N~~@~~wF~~_N~~?N~~?F~~_@~~w?N~~??~~{?@~~w?@~~w??~~{??N~~??@~~w??F~~_??N~~???N~~???F~~_??@~~w???N~~????~~{???@~~w???@~~w????~~{????N~~????@~~w????F~~_????N~~?????N~~?????F~~_????@~~w?????N~~??????~~{?????@~~w?????@~~w??????~~{??????N~~??????@~~w??????F~~_??????N~~???????N~~???????F~~_??????@~~{???????N~~o???????~~~_??????@~~~_??????@~~~o???????~~~{???????N~~~_??????@~~~}???????F~~~{???????N~~~{???????N~~~}???????F~~~~_??????@~~~~{???????N~~~~o???????~~~~~_??????@~~~~~_??????@~~w',
8666
'C80Pow16': '~?@O~~~~~~~~~~~~~~~~~~~~~~|~~x~~w~~{N~~@~~wF~~_N~~?N~~?F~~_@~~w?N~~??~~{?@~~w?@~~w??~~{??N~~??@~~w??F~~_??N~~???N~~???F~~_??@~~w???N~~????~~{???@~~w???@~~w????~~{????N~~????@~~w????F~~_????N~~?????N~~?????F~~_????@~~w?????N~~??????~~{?????@~~w?????@~~w??????~~{??????N~~??????@~~w??????F~~_??????N~~???????N~~???????F~~_??????@~~w???????N~~_???????~~~???????@~~~???????@~~~_???????~~~w???????N~~~???????@~~~{???????F~~~w???????N~~~w???????N~~~{???????F~~~~???????@~~~~w???????N~~~~_???????~~~~~???????@~~~~~???????@~~~~~_???????~~{',
8667
'C81Pow16': '~?@P~~~~~~~~~~~~~~~~~~~~~~|~~x~~w~~{N~~@~~wF~~_N~~?N~~?F~~_@~~w?N~~??~~{?@~~w?@~~w??~~{??N~~??@~~w??F~~_??N~~???N~~???F~~_??@~~w???N~~????~~{???@~~w???@~~w????~~{????N~~????@~~w????F~~_????N~~?????N~~?????F~~_????@~~w?????N~~??????~~{?????@~~w?????@~~w??????~~{??????N~~??????@~~w??????F~~_??????N~~???????N~~???????F~~_??????@~~w???????N~~????????~~}???????@~~}???????@~~~????????~~~o???????N~~}???????@~~~w???????F~~~o???????N~~~o???????N~~~w???????F~~~}???????@~~~~o???????N~~~~????????~~~~}???????@~~~~}???????@~~~~~????????~~~~~o???????N~~',
8668
'C82Pow16': '~?@Q~~~~~~~~~~~~~~~~~~~~~~|~~x~~w~~{N~~@~~wF~~_N~~?N~~?F~~_@~~w?N~~??~~{?@~~w?@~~w??~~{??N~~??@~~w??F~~_??N~~???N~~???F~~_??@~~w???N~~????~~{???@~~w???@~~w????~~{????N~~????@~~w????F~~_????N~~?????N~~?????F~~_????@~~w?????N~~??????~~{?????@~~w?????@~~w??????~~{??????N~~??????@~~w??????F~~_??????N~~???????N~~???????F~~_??????@~~w???????N~~????????~~{???????@~~{???????@~~}????????~~~_???????N~~{???????@~~~o???????F~~~_???????N~~~_???????N~~~o???????F~~~{???????@~~~~_???????N~~~}????????~~~~{???????@~~~~{???????@~~~~}????????~~~~~_???????N~~~~{???????@~~w',
8669
'C83Pow16': '~?@R~~~~~~~~~~~~~~~~~~~~~~|~~x~~w~~{N~~@~~wF~~_N~~?N~~?F~~_@~~w?N~~??~~{?@~~w?@~~w??~~{??N~~??@~~w??F~~_??N~~???N~~???F~~_??@~~w???N~~????~~{???@~~w???@~~w????~~{????N~~????@~~w????F~~_????N~~?????N~~?????F~~_????@~~w?????N~~??????~~{?????@~~w?????@~~w??????~~{??????N~~??????@~~w??????F~~_??????N~~???????N~~???????F~~_??????@~~w???????N~~????????~~{???????@~~w???????@~~{????????~~~????????N~~w???????@~~~_???????F~~~????????N~~~????????N~~~_???????F~~~w???????@~~~~????????N~~~{????????~~~~w???????@~~~~w???????@~~~~{????????~~~~~????????N~~~~w???????@~~~~~_???????F~~_',
8670
'C84Pow16': '~?@S~~~~~~~~~~~~~~~~~~~~~~|~~x~~w~~{N~~@~~wF~~_N~~?N~~?F~~_@~~w?N~~??~~{?@~~w?@~~w??~~{??N~~??@~~w??F~~_??N~~???N~~???F~~_??@~~w???N~~????~~{???@~~w???@~~w????~~{????N~~????@~~w????F~~_????N~~?????N~~?????F~~_????@~~w?????N~~??????~~{?????@~~w?????@~~w??????~~{??????N~~??????@~~w??????F~~_??????N~~???????N~~???????F~~_??????@~~w???????N~~????????~~{???????@~~w???????@~~w????????~~}????????N~~o???????@~~~????????F~~}????????N~~}????????N~~~????????F~~~o???????@~~~}????????N~~~w????????~~~~o???????@~~~~o???????@~~~~w????????~~~~}????????N~~~~o???????@~~~~~????????F~~~~}????????N~~',
8671
'C85Pow16': '~?@T~~~~~~~~~~~~~~~~~~~~~~|~~x~~w~~{N~~@~~wF~~_N~~?N~~?F~~_@~~w?N~~??~~{?@~~w?@~~w??~~{??N~~??@~~w??F~~_??N~~???N~~???F~~_??@~~w???N~~????~~{???@~~w???@~~w????~~{????N~~????@~~w????F~~_????N~~?????N~~?????F~~_????@~~w?????N~~??????~~{?????@~~w?????@~~w??????~~{??????N~~??????@~~w??????F~~_??????N~~???????N~~???????F~~_??????@~~w???????N~~????????~~{???????@~~w???????@~~w????????~~{????????N~~_???????@~~}????????F~~{????????N~~{????????N~~}????????F~~~_???????@~~~{????????N~~~o????????~~~~_???????@~~~~_???????@~~~~o????????~~~~{????????N~~~~_???????@~~~~}????????F~~~~{????????N~~~~{????????N~~',
8672
'C86Pow16': '~?@U~~~~~~~~~~~~~~~~~~~~~~|~~x~~w~~{N~~@~~wF~~_N~~?N~~?F~~_@~~w?N~~??~~{?@~~w?@~~w??~~{??N~~??@~~w??F~~_??N~~???N~~???F~~_??@~~w???N~~????~~{???@~~w???@~~w????~~{????N~~????@~~w????F~~_????N~~?????N~~?????F~~_????@~~w?????N~~??????~~{?????@~~w?????@~~w??????~~{??????N~~??????@~~w??????F~~_??????N~~???????N~~???????F~~_??????@~~w???????N~~????????~~{???????@~~w???????@~~w????????~~{????????N~~????????@~~{????????F~~w????????N~~w????????N~~{????????F~~~????????@~~~w????????N~~~_????????~~~~????????@~~~~????????@~~~~_????????~~~~w????????N~~~~????????@~~~~{????????F~~~~w????????N~~~~w????????N~~~~{????????F~~_',
8673
'C87Pow16': '~?@V~~~~~~~~~~~~~~~~~~~~~~|~~x~~w~~{N~~@~~wF~~_N~~?N~~?F~~_@~~w?N~~??~~{?@~~w?@~~w??~~{??N~~??@~~w??F~~_??N~~???N~~???F~~_??@~~w???N~~????~~{???@~~w???@~~w????~~{????N~~????@~~w????F~~_????N~~?????N~~?????F~~_????@~~w?????N~~??????~~{?????@~~w?????@~~w??????~~{??????N~~??????@~~w??????F~~_??????N~~???????N~~???????F~~_??????@~~w???????N~~????????~~{???????@~~w???????@~~w????????~~{????????N~~????????@~~w????????F~~o????????N~~o????????N~~w????????F~~}????????@~~~o????????N~~~?????????~~~}????????@~~~}????????@~~~~?????????~~~~o????????N~~~}????????@~~~~w????????F~~~~o????????N~~~~o????????N~~~~w????????F~~~~}????????@~~w',
8674
'C88Pow16': '~?@W~~~~~~~~~~~~~~~~~~~~~~|~~x~~w~~{N~~@~~wF~~_N~~?N~~?F~~_@~~w?N~~??~~{?@~~w?@~~w??~~{??N~~??@~~w??F~~_??N~~???N~~???F~~_??@~~w???N~~????~~{???@~~w???@~~w????~~{????N~~????@~~w????F~~_????N~~?????N~~?????F~~_????@~~w?????N~~??????~~{?????@~~w?????@~~w??????~~{??????N~~??????@~~w??????F~~_??????N~~???????N~~???????F~~_??????@~~w???????N~~????????~~{???????@~~w???????@~~w????????~~{????????N~~????????@~~w????????F~~_????????N~~_????????N~~o????????F~~{????????@~~~_????????N~~}?????????~~~{????????@~~~{????????@~~~}?????????~~~~_????????N~~~{????????@~~~~o????????F~~~~_????????N~~~~_????????N~~~~o????????F~~~~{????????@~~~~~_????????N~~',
8675
'C89Pow16': '~?@X~~~~~~~~~~~~~~~~~~~~~~|~~x~~w~~{N~~@~~wF~~_N~~?N~~?F~~_@~~w?N~~??~~{?@~~w?@~~w??~~{??N~~??@~~w??F~~_??N~~???N~~???F~~_??@~~w???N~~????~~{???@~~w???@~~w????~~{????N~~????@~~w????F~~_????N~~?????N~~?????F~~_????@~~w?????N~~??????~~{?????@~~w?????@~~w??????~~{??????N~~??????@~~w??????F~~_??????N~~???????N~~???????F~~_??????@~~w???????N~~????????~~{???????@~~w???????@~~w????????~~{????????N~~????????@~~w????????F~~_????????N~~?????????N~~_????????F~~w????????@~~~?????????N~~{?????????~~~w????????@~~~w????????@~~~{?????????~~~~?????????N~~~w????????@~~~~_????????F~~~~?????????N~~~~?????????N~~~~_????????F~~~~w????????@~~~~~?????????N~~~~{?????????~~{',
8676
'C90Pow16': '~?@Y~~~~~~~~~~~~~~~~~~~~~~|~~x~~w~~{N~~@~~wF~~_N~~?N~~?F~~_@~~w?N~~??~~{?@~~w?@~~w??~~{??N~~??@~~w??F~~_??N~~???N~~???F~~_??@~~w???N~~????~~{???@~~w???@~~w????~~{????N~~????@~~w????F~~_????N~~?????N~~?????F~~_????@~~w?????N~~??????~~{?????@~~w?????@~~w??????~~{??????N~~??????@~~w??????F~~_??????N~~???????N~~???????F~~_??????@~~w???????N~~????????~~{???????@~~w???????@~~w????????~~{????????N~~????????@~~w????????F~~_????????N~~?????????N~~?????????F~~o????????@~~}?????????N~~w?????????~~~o????????@~~~o????????@~~~w?????????~~~}?????????N~~~o????????@~~~~?????????F~~~}?????????N~~~}?????????N~~~~?????????F~~~~o????????@~~~~}?????????N~~~~w?????????~~~~~o????????@~~w',
8677
'C91Pow16': '~?@Z~~~~~~~~~~~~~~~~~~~~~~|~~x~~w~~{N~~@~~wF~~_N~~?N~~?F~~_@~~w?N~~??~~{?@~~w?@~~w??~~{??N~~??@~~w??F~~_??N~~???N~~???F~~_??@~~w???N~~????~~{???@~~w???@~~w????~~{????N~~????@~~w????F~~_????N~~?????N~~?????F~~_????@~~w?????N~~??????~~{?????@~~w?????@~~w??????~~{??????N~~??????@~~w??????F~~_??????N~~???????N~~???????F~~_??????@~~w???????N~~????????~~{???????@~~w???????@~~w????????~~{????????N~~????????@~~w????????F~~_????????N~~?????????N~~?????????F~~_????????@~~{?????????N~~o?????????~~~_????????@~~~_????????@~~~o?????????~~~{?????????N~~~_????????@~~~}?????????F~~~{?????????N~~~{?????????N~~~}?????????F~~~~_????????@~~~~{?????????N~~~~o?????????~~~~~_????????@~~~~~_????????@~~w',
8678
'C92Pow16': '~?@[~~~~~~~~~~~~~~~~~~~~~~|~~x~~w~~{N~~@~~wF~~_N~~?N~~?F~~_@~~w?N~~??~~{?@~~w?@~~w??~~{??N~~??@~~w??F~~_??N~~???N~~???F~~_??@~~w???N~~????~~{???@~~w???@~~w????~~{????N~~????@~~w????F~~_????N~~?????N~~?????F~~_????@~~w?????N~~??????~~{?????@~~w?????@~~w??????~~{??????N~~??????@~~w??????F~~_??????N~~???????N~~???????F~~_??????@~~w???????N~~????????~~{???????@~~w???????@~~w????????~~{????????N~~????????@~~w????????F~~_????????N~~?????????N~~?????????F~~_????????@~~w?????????N~~_?????????~~~?????????@~~~?????????@~~~_?????????~~~w?????????N~~~?????????@~~~{?????????F~~~w?????????N~~~w?????????N~~~{?????????F~~~~?????????@~~~~w?????????N~~~~_?????????~~~~~?????????@~~~~~?????????@~~~~~_?????????~~{',
8679
'C93Pow16': '~?@\\~~~~~~~~~~~~~~~~~~~~~~|~~x~~w~~{N~~@~~wF~~_N~~?N~~?F~~_@~~w?N~~??~~{?@~~w?@~~w??~~{??N~~??@~~w??F~~_??N~~???N~~???F~~_??@~~w???N~~????~~{???@~~w???@~~w????~~{????N~~????@~~w????F~~_????N~~?????N~~?????F~~_????@~~w?????N~~??????~~{?????@~~w?????@~~w??????~~{??????N~~??????@~~w??????F~~_??????N~~???????N~~???????F~~_??????@~~w???????N~~????????~~{???????@~~w???????@~~w????????~~{????????N~~????????@~~w????????F~~_????????N~~?????????N~~?????????F~~_????????@~~w?????????N~~??????????~~}?????????@~~}?????????@~~~??????????~~~o?????????N~~}?????????@~~~w?????????F~~~o?????????N~~~o?????????N~~~w?????????F~~~}?????????@~~~~o?????????N~~~~??????????~~~~}?????????@~~~~}?????????@~~~~~??????????~~~~~o?????????N~~',
8680
'C94Pow16': '~?@]~~~~~~~~~~~~~~~~~~~~~~|~~x~~w~~{N~~@~~wF~~_N~~?N~~?F~~_@~~w?N~~??~~{?@~~w?@~~w??~~{??N~~??@~~w??F~~_??N~~???N~~???F~~_??@~~w???N~~????~~{???@~~w???@~~w????~~{????N~~????@~~w????F~~_????N~~?????N~~?????F~~_????@~~w?????N~~??????~~{?????@~~w?????@~~w??????~~{??????N~~??????@~~w??????F~~_??????N~~???????N~~???????F~~_??????@~~w???????N~~????????~~{???????@~~w???????@~~w????????~~{????????N~~????????@~~w????????F~~_????????N~~?????????N~~?????????F~~_????????@~~w?????????N~~??????????~~{?????????@~~{?????????@~~}??????????~~~_?????????N~~{?????????@~~~o?????????F~~~_?????????N~~~_?????????N~~~o?????????F~~~{?????????@~~~~_?????????N~~~}??????????~~~~{?????????@~~~~{?????????@~~~~}??????????~~~~~_?????????N~~~~{?????????@~~w',
8681
'C95Pow16': '~?@^~~~~~~~~~~~~~~~~~~~~~~|~~x~~w~~{N~~@~~wF~~_N~~?N~~?F~~_@~~w?N~~??~~{?@~~w?@~~w??~~{??N~~??@~~w??F~~_??N~~???N~~???F~~_??@~~w???N~~????~~{???@~~w???@~~w????~~{????N~~????@~~w????F~~_????N~~?????N~~?????F~~_????@~~w?????N~~??????~~{?????@~~w?????@~~w??????~~{??????N~~??????@~~w??????F~~_??????N~~???????N~~???????F~~_??????@~~w???????N~~????????~~{???????@~~w???????@~~w????????~~{????????N~~????????@~~w????????F~~_????????N~~?????????N~~?????????F~~_????????@~~w?????????N~~??????????~~{?????????@~~w?????????@~~{??????????~~~??????????N~~w?????????@~~~_?????????F~~~??????????N~~~??????????N~~~_?????????F~~~w?????????@~~~~??????????N~~~{??????????~~~~w?????????@~~~~w?????????@~~~~{??????????~~~~~??????????N~~~~w?????????@~~~~~_?????????F~~_',
8682
'C96Pow16': '~?@_~~~~~~~~~~~~~~~~~~~~~~|~~x~~w~~{N~~@~~wF~~_N~~?N~~?F~~_@~~w?N~~??~~{?@~~w?@~~w??~~{??N~~??@~~w??F~~_??N~~???N~~???F~~_??@~~w???N~~????~~{???@~~w???@~~w????~~{????N~~????@~~w????F~~_????N~~?????N~~?????F~~_????@~~w?????N~~??????~~{?????@~~w?????@~~w??????~~{??????N~~??????@~~w??????F~~_??????N~~???????N~~???????F~~_??????@~~w???????N~~????????~~{???????@~~w???????@~~w????????~~{????????N~~????????@~~w????????F~~_????????N~~?????????N~~?????????F~~_????????@~~w?????????N~~??????????~~{?????????@~~w?????????@~~w??????????~~}??????????N~~o?????????@~~~??????????F~~}??????????N~~}??????????N~~~??????????F~~~o?????????@~~~}??????????N~~~w??????????~~~~o?????????@~~~~o?????????@~~~~w??????????~~~~}??????????N~~~~o?????????@~~~~~??????????F~~~~}??????????N~~',
8683
'C97Pow16': '~?@`~~~~~~~~~~~~~~~~~~~~~~|~~x~~w~~{N~~@~~wF~~_N~~?N~~?F~~_@~~w?N~~??~~{?@~~w?@~~w??~~{??N~~??@~~w??F~~_??N~~???N~~???F~~_??@~~w???N~~????~~{???@~~w???@~~w????~~{????N~~????@~~w????F~~_????N~~?????N~~?????F~~_????@~~w?????N~~??????~~{?????@~~w?????@~~w??????~~{??????N~~??????@~~w??????F~~_??????N~~???????N~~???????F~~_??????@~~w???????N~~????????~~{???????@~~w???????@~~w????????~~{????????N~~????????@~~w????????F~~_????????N~~?????????N~~?????????F~~_????????@~~w?????????N~~??????????~~{?????????@~~w?????????@~~w??????????~~{??????????N~~_?????????@~~}??????????F~~{??????????N~~{??????????N~~}??????????F~~~_?????????@~~~{??????????N~~~o??????????~~~~_?????????@~~~~_?????????@~~~~o??????????~~~~{??????????N~~~~_?????????@~~~~}??????????F~~~~{??????????N~~~~{??????????N~~',
8684
'C98Pow16': '~?@a~~~~~~~~~~~~~~~~~~~~~~|~~x~~w~~{N~~@~~wF~~_N~~?N~~?F~~_@~~w?N~~??~~{?@~~w?@~~w??~~{??N~~??@~~w??F~~_??N~~???N~~???F~~_??@~~w???N~~????~~{???@~~w???@~~w????~~{????N~~????@~~w????F~~_????N~~?????N~~?????F~~_????@~~w?????N~~??????~~{?????@~~w?????@~~w??????~~{??????N~~??????@~~w??????F~~_??????N~~???????N~~???????F~~_??????@~~w???????N~~????????~~{???????@~~w???????@~~w????????~~{????????N~~????????@~~w????????F~~_????????N~~?????????N~~?????????F~~_????????@~~w?????????N~~??????????~~{?????????@~~w?????????@~~w??????????~~{??????????N~~??????????@~~{??????????F~~w??????????N~~w??????????N~~{??????????F~~~??????????@~~~w??????????N~~~_??????????~~~~??????????@~~~~??????????@~~~~_??????????~~~~w??????????N~~~~??????????@~~~~{??????????F~~~~w??????????N~~~~w??????????N~~~~{??????????F~~_',
8685
'C99Pow16': '~?@b~~~~~~~~~~~~~~~~~~~~~~|~~x~~w~~{N~~@~~wF~~_N~~?N~~?F~~_@~~w?N~~??~~{?@~~w?@~~w??~~{??N~~??@~~w??F~~_??N~~???N~~???F~~_??@~~w???N~~????~~{???@~~w???@~~w????~~{????N~~????@~~w????F~~_????N~~?????N~~?????F~~_????@~~w?????N~~??????~~{?????@~~w?????@~~w??????~~{??????N~~??????@~~w??????F~~_??????N~~???????N~~???????F~~_??????@~~w???????N~~????????~~{???????@~~w???????@~~w????????~~{????????N~~????????@~~w????????F~~_????????N~~?????????N~~?????????F~~_????????@~~w?????????N~~??????????~~{?????????@~~w?????????@~~w??????????~~{??????????N~~??????????@~~w??????????F~~o??????????N~~o??????????N~~w??????????F~~}??????????@~~~o??????????N~~~???????????~~~}??????????@~~~}??????????@~~~~???????????~~~~o??????????N~~~}??????????@~~~~w??????????F~~~~o??????????N~~~~o??????????N~~~~w??????????F~~~~}??????????@~~w',
8686
'DenBip04': 'G?vtb_',
8687
'DenBip05': 'I?B^fPwm?',
8688
'DenBip06': 'K??BjzgnFw]?',
8689
'DenBip07': 'M???FfqNf{\\of_~??',
8690
'DenBip08': 'O????Bl~BM]Ohou_ZGC}?',
8691
'DenBip09': 'Q??????~m}TknozwVKBf?]gBe_?',
8692
'DenBip10': 'S???????F}{[v_rKFwDJ_mKBPo@^_Bl??',
8693
'DenBip11': 'U?????????H}v][u^rA^OjeBMgFf?Dz_@sw?m[??',
8694
'DenBip12': 'W???????????}mh]UReyGJZBG{FuGBiW@fo?fv?E~o?E}??',
8695
'DenBip13': 'Y?????????????\\{u{q^gznb[uDw{FYS@V[?I~?Fqc?]XO?}J_?bn???',
8696
'DenBip14': '[???????????????\\l}Lk^sOt]DW{@\\iBVL?YVoE|C?J\\G?kho?k}??SZg?Er]??',
8697
'DenBip15': ']?????????????????@bv{Hzvgdy~E\\mBJr_JZgBx|?P|o?xUo?Rv_?Uww?A}~??mnW?BY{???',
8698
'DenBip16': '_????????????????????s]V\\ndXs`rR`YmO}fOC}T?Jy]?wnC?jHw?VmG?Am|??{|{?Bng_?FM~_?F@^_??',
8699
'DenBip17': 'a??????????????????????@}k~^JtUdoTzW|hKEp^?E}Y?zRy?F\\q?^~N?Drp_?Hj{?BiVG?EUko?Apz_?BL^G??S~e???',
8700
'DenBip18': 'c?????????????????????????FZ{^dqQ^foW}iDkTgU|__]z]?om]?PP|_Ahuo?LMm?B|Vo?DYmG?EyzW?BkBk??Ufd??DFhg??H]F_??',
8701
'DenBip19': 'e????????????????????????????B_v|DlmlRsfF\\cFNx?y\\F_va~?RZkODWNk?ulh??vsk?D^_k?@uFw?@unE??T}c_?FJbo??X~No??vlR_??nIf_???',
8702
'DenBip20': 'g???????????????????????????????Bw[s}Y`ufbYUt\\oUF{_glyOTwsWAjTq?EVfOAYZe?FuLm??^S}?Bv|Q??plx??FXyc??@uZg??Z^R???lLN_??Kw\\W??@Avy???',
8703
'DenBip21': 'i???????????????????????????????????bw]ihi|]{y?wm~OrRdWLIvkDdvS?M\\Tw?|Kn_EuX]?Byei?B]p~_?dfcw?Eu|N??Y}hC??jiJg??}YqO??V\\XC??@itV???Zzk???@N\\u_???',
8704
'DenBip22': 'k??????????????????????????????????????DZ`{k`\\]x}HCa^tCMqB{AAN|_LFUw?xLto?\\~L_AF|d?AYm}O?zhps?FjaR??QVwq??V}ig??szBW??^q]{??F\\Xz_??~bQc???zP|???DZsb???BesU_???',
8705
'DenBip23': 'm??????????????????????????????????????????N{wVWe~MTt\\a[{xHDte{?L}dIBXXf_BUJqoCglzO@]ikw?vfm[?@VW}o?Eij]??NIza??p}TE??WxH}??CrZj???^Bbs??Ae}qW??EPxr_??FP\\\\???APVzG???~[~}????',
8706
'DenBip24': 'o??????????????????????????????????????????????fmo^|cvGJ[wwaFFMw^rJu@z|C_@zITWEIY~?AQyLw?zUf|?@itf_?\\?|v??O|MZ??rJxv??\\Smh??Cht}g??meKt??@aNXs??BJ`bw??@rroo??An^Jk???fvfF???E|r_O???]gvo_???',
8707
'DenBip25': 'q??????????????????????????????????????????????????VyFWdNXVU_A~{M~eqa}UXeDX]s{F~@ng@VvQW?oxZT_@I|Z[?NR]K_?XrjM??MLZz_?Vuyo??AUyk{??VR{p??Bqmnw??Du[Mg??A{jLc??B\\b`q???t[[l???FirH{???THnhO???xmun????_m}b_????',
8708
'DenBip26': 's??????????????????????????????????????????????????????YC^ZVd|jqjzOSpYzNCF{Z}yBEFnc@U^gM?VBfr_FdrFO?S{yf_?[p`lo?aoN|_?Czvyg?DqzAY??HE]mo?@kn{V??CRg^k??EnwXE??AoZ{r???B[tj_??F~CNI???Xkp{G???NF[MO???M}WqO???JMbJW???CY{c}????',
8709
'DenBip27': 'u??????????????????????????????????????????????????????????CvoFy\\|m]rvHdQdIvVcEX~GXBjjFP?mLVdo@}OzZ?Fn\\tS?ZC~hO?oukvG?PYkN[?CnBy[??wL{jG?@xftU_?A|Xbt??FhMcu??Am^yk_??FzNGw??C~k?z???TMtbg???Tjm[g???RMUzO???OfXs{???B}l~v????{B~ro???BJTEr_????',
8710
'DenBip28': 'w???????????????????????????????????????????????????????????????mXyXEvO\\JUYLy^bYEJz`MR}ioP~YNWAhfVQ_ANi~e?khtkg?NbT~C?KVm]O?DT\\S^??csnjS?Aieqso?BXZYh??ApL~E??AeyP\\_??NyhNo??Fzcct???Ujz^M???Tu[P[???ymDpK???J~@IY???Bwxw^????mRqko???BZ~?UO???ELRX\\_???FQjme_????',
8711
'DenBip29': 'y???????????????????????????????????????????????????????????????????@IMztgx}zdFy?lLorZyR_{oh~UA}pI]O[m~n|?W}qqY?jfONM?Bm~[u?FYKr\\??HUd}k?B_}uPW?@dTMz_?F|BL~o?AceunG??s@}zc??DvyWa_??EYNj^???fuakM???ztotC???DNgU|???E~[^eo???mZmFM????Mj}NG???F|[rS_???Efvjmo???BrXgLg????tnZoA?????',
8712
'DenBip30': '{????????????????????????????????????????????????????????????????????????@M~gDWVnRFRIj]p[kFmXhAxUNx_H^MlK?~yJld?^P[{M?RqA|x?ENJq}??~]wej?@mRjS[?E@s|io?DIiUZW?@vWzWs??vn]]M??BuU[t???W\\yEL_??JbEns???k~`Gt???Oktk{_??BfMXWo???ZBXvT???AjuPvk???DwQ}}W???C~ckBo???@`uzN[????UtHil????Da^I[w????]Ybz_?????',
8713
'DenBip31': '}?????????????????????????????????????????????????????????????????????????????D~Zo\\TmRxXWs|s~]fb]HMc?b|vV?JSyo^_~xvup?QB[rvOBJOZ\\k?bPJw~?AHmsva?CykJn_?B|Xeps?Bz}zz[??z~ZPm_?A{~LCW??YzIZ`_??l]\\Zx_??WI^|X_??Kibe|o??@dUUyk???Pueg}_??BruA|a???Btte^[???CslJg{????Yf}qg????q{WVu????@xYyak????^bb}K_????cxirR_????xiuj`??????',
8714
'DenBip32': '~?@???????????????????????????????????????????????????????????????????????????????????Bq^AaZYkUrZ`QMie}Ufcquo}Sh^n_Dq[^aoR}[IwGFnjBw{?jbu^Q?@EhmmU?F]wkLq??nMuRa?BDd{Zg??H^Jlt_??KE|\\]??ZjTZlg??U{Yrq_??|iuaY???Ni]NMG??Eane`e???|ZZhko??@R^XgF???A|wmiu???C}T|}Y???Agz[LT????CXnyX_???@pyjci????^KOffO????D|}{|o????iFzO^_????Lp_Utw????EY[\\qa?????',
8715
'DenBip33': '~?@A????????????????????????????????????????????????????????????????????????????????????????qHWw~jKt{^YRwTytOU}VP{gvstuK_VpzQQoDzTZm_?ZTzKwG?E|zI~?BCB}t~?EA^V}A?AmS`~V_?FmqBMw?Aest|B??Cb|vPc??^\\V@UW??}z|jPw??OKjmNw??Bc}ZNZ???|LH}rG??AZdShy_??F|\\FRS???EzoVxn???Auhm?^_???iKujt_???@Nl`lZ????^MafBc????rut_Z_????Vij|`?????AzApx{????ApmQ]u?????hZLT|O????ApVxpZ??????',
8716
'DenBip34': '~?@C?????????????????????????????????????????????????????????????????????????????????????????????Elof_{s{v`\\g[}R}}WcftkSw\\HnepcFMI\\to_XZvils?ncbfUODbsEbu_EzoUJt?Bw@vygo?fMZSEk?A[NtPM_?TFmmDe??vUaht_??EX}\\ww??^W]HEe???wK}fN???Z_uNck??@J{K\\Yo??B^hOW^???@h|y\\A????iE}K^_???c}VNqC???ErHXFe_???SCgzvk????\\QSftS????zm}pI{????]RSy}o????CtcVYJ_????|}gNyO????AEm`^LO????@mnRbR?????BP}mML??????',
8717
'DenBip35': '~?@E???????????????????????????????????????????????????????????????????????????????????????????????????@~?{ytjoqK{y^tWY|G^tXATpAuVQ~v_`~dvwG@S{AJ^wDUPVUpoFYrlGroAFFHrLw?c]mvhG?FBljXf_?JNLoNL??`J[nxa??MhDvLe??Vu}dVW??DxUVzgO??Dl|_pm??Bq|NudW??DZfzbu???ErFgp^_??BNVXXIG???vt|kv}???Bcs{[z????MDjlta????X^Ioim????JbflUE????WN^VAq????F}cV]Y_????H^ZPa]????@pW}bRw????AtF}~No????E|ZygQo????BXce{~O?????IbbZ\\i??????',
8718
'DenBip36': '~?@G?????????????????????????????????????????????????????????????????????????????????????????????????????????S|ip|PoRN[[mHRhwvyD{TepAwL{Wyx`AzgjxxoCrKPivwB[mb][?@F}kSrk?LdVga}?Aqd^cxO?^]ssgf_?tT^knb??{NeJ@l??Vc~R``??@{vM{v???_pkb|r??BkwDd}c??FPJNfT_??BQM}Edg??A~@^DhO???Yi^f]^????j}c^KG???AcJVyZ_???`YYuNY????tMZMQu?????vqJqf_???FzyhR}W????MUw]j\\????@uvov[W????CMKJtew????F^f]|\\G?????in[VwG?????qrP[|K?????AAXyq^o?????ZzBAQm_?????',
8719
'DenBip37': '~?@I???????????????????????????????????????????????????????????????????????????????????????????????????????????????ytfFeq]Jp[pbdpVMZx?kaam]ybp||^r}Bnuph|wEAE}tMsBuvM[|a?SSuJ\\y_E|Ir}J[??xv{\\c_?J~}x}v??mPtmIw_?NxCuDtO?D{}Jw@g??B^y}b@_?BR}QOq{??D[g|Fc[??B\\xA]mw???]Hpl^S???\\]N}qF_??AZ^NrOC???TT^pEUO???xRbN_z????sb[|hz_???PVelKzo???@kzKTro????][Uf\\C_???AjXvDsE????EMLcMxw????EzMS|Us?????gzf^Fo?????Y{Qkrr?????FulG[qc?????Jc~jJM_?????FILX{m_?????z[Kx|L???????',
8720
'DenBip38': '~?@K?????????????????????????????????????????????????????????????????????????????????????????????????????????????????????]pmC]lAbMh\\{ETL{vJg`|OC\\mtE{yXJQcBPur[|w?LY^d]E?]g}zTv_CjDuhbs?NMwH]mw?zyZLWGO?w`vwHJo?V}[qB}_?CPNlTtI??OnjUe{O?@|Ao`{n??DBzLGmi??C{}RkaK??@JCle}x???i|faxuO??BHtv^NU???Q]TeXQw???PZNraRo???o}C~O}_???^RcxTxg???FfIKkP{????fzRkPF????@F{gZOv????@\\RFuPe????CQ\\ZgzS?????]u\\ZiC?????mG[B|VO????B|Qz|zO?????TAmY~pW?????h~Lae}_?????d}Hm}}_?????Z_~YGhg?????BL]Hb]E??????',
8721
'DenBip39': '~?@M????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????\\z{uF`ZfBQJRzG\\kldU`drkY]iE|?\\suJBsXdyq]?e{fjPhwDVhsS`|?Rxu{Keg?[_Es}vo?bmv]VjO?M[ID]Vs?Awpk^uF??NrPVuIG?BA~lv~v_?CE|sclV??DjFya{m???~u?ZoZ_??r^LdjQO??@VbxGW~???J}\\EcFK???h[v|KoG???~RvBK^G???\\TO~XKg???Bt^fLiW????{RwKv}w????CnJ]Kz_???DQk}epd????@Y~FcMh????@jeb{~e?????unrx@@o????Eg\\NXRv?????NoPI^|C?????TP]^lIG?????Qvuc~z??????QCepfnw?????AJbRZuu??????lQI^v{_?????ByExvah???????',
8722
'DenBip40': '~?@O??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????HfOfxuz}oR~sesr`\\rfhcMiojxpa~br?rXoUHnfRWSCDey}DY_Gb\\[wjq?NWnG{TS?L[bwn@k?Tov`FVS?B{TzkxM??{V]WiFO??KTsl\\NO?F~RHJIa_?Anx_}`d??@ZRZjj\\o??Fnp}M}s??Du~kCYI???]hj]xRW???nq`YY}g???ewBPxs{???LMHPY|U???FU|]lL^_???pIltU^g???@Qd`e^to???B|~Wxw]????@b]Xk]S_???ABlzijEo????uJYlti_????@vReS\\p?????AJnVpdo?????PDi]zxO?????aicuSx{?????W|orEXY?????F]U\\LGh??????|HdvTkw?????BKouh{UO?????BQj`yzD??????DA{]Hfb_??????',
8723
'DenBip41': '~?@Q?????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????|c[lz^~VNspxGCcz~LEb_kY`tF|W|prEkecBZhNo\\Y?\\e]e\\DV?y^U{pJ}?K\\~txUk?Y]UONTr?ENEJLL|o?fYajNsW?@rMbVm~w?CMWnWZf_?E^|ZyQ\\O??]ehRXxo??fdVLuaE??AMzMVrDo??X~]EoEb???d{{eYge???tFwa^Sq???LSANvkt???BWmRiXm_???KMzWs~u???BxO^heeg???@~~sYv_????Ex]bKmZ_???Bxq{~o@?????Lo~elTo????DmykZu^?????LpIN~~k?????JIFUK~e?????{`FncT[?????^pWOi|L?????BN`^PmPO?????\\eNUoDu?????AtKNwMxO?????ARVTRjlO?????ConKK}}??????AhMPusYw??????b^Q]zzI???????',
8724
'DenBip42': '~?@S???????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????@sY~wPwMHZLJpkWnzKysJWbsD~owk@JHBlnaw^Q^q?ro_x[wz`Q[?~y_er[m?YxipIh^_@wp~tmi_?y`x[Yra?@dUwyw~{??x\\VbjX_?EbSr[zGg?AR[}v_r???~LmVYuc??Bg_yNygw??^dxljEl_??Va}F|iR???JzF{YbO???GfXQ\\te_??FY~aJu]G???MnxxDVA???B[xtAMXc????BJQ|Fyw???CTj~[DfO????^oYw}A{????\\k{gZO{????@`Daz\\vo????^OYxztu?????BvA{rBm??????^]UMMl?????UYJprYB_????AnxwMB]W?????yvC`mwL?????AzntLATk?????DiDzWnBG??????@l[Kx}w?????BRAEZpVs??????~yS{`PX??????EcY\\l^zO??????WMO{mY|???????',
8725
'DenBip43': '~?@U??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????Ea[yJ_~Y^IJ{IT[_a^jpp]fpqcartoN}LrTXk?FOJld^i_`xymraK_AEz[t[boFW\\XUyTk?nzMgEwt?Ac~bNadW?@~mURIzk?@^hX[XQk?AqfXmHq[??QL{F^xO_?E`ZfJkdW??^bIgE}F_??l~~PXDM???\\y`Jftn???[YWN^}oo??F`FzZ@}G???RrVwwjx_??@c]]UP^E???FexM|Tas???F]N]~lR_????zUVe|]s????DNn_y]r_???EV?lwle[????E_~NKjkO????n{kdmIH?????YnBewJq_????UzwPRjko????CqvnlfEo?????sAe^Q{n?????@xf`xiQq?????AHJPbv^o?????B@ZI{~TS?????AnQ@W}Y{??????Bl|[\\@[_?????BcxmiDe[??????^~QHRcpO??????UC`Pv}f_??????r]YmRRO_???????',
8726
'DenBip44': '~?@W?????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????BXUVkYxN\\b`FwYV``xm]U|]JynkLYx?S[]krVt?nmTNdAMoSxkffYq?BHn{Cc]s?icyD\\sqo@{z{mQ\\p?C{~EF`Xg?@y|aUqNG?BMn{ynaB??SiKS]zV_?DbtvmG@U??K]uy|iSG??c\\nDf]`???ofZ_Fm^???Hj\\nTUwo??@yLUn|la???|mTfKbqo??@N_RIvbm???CCku|Hte???DWfPo|cm????{fav@za????xXfN\\\\Co????BJjrWq}????VagMdhFw????Dyv~kYb?????VtGpwqwo????ZYD\\[yFW????CdIrR}rA?????wZv?pjHo????A@aPd}Nz?????EzSZaxHY?????D~vCJtyq?????BbjjWyvl??????m`B}hwQo?????BV{L|MXg??????WzH`jZh_??????}a@hON~_??????iF[Wep^O??????NZXpqnu_???????Isf]jie???????',
8727
'DenBip45': '~?@Y?????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????TpwUmJmH[U|rtk@?Nl~B\\[CzXWKsjio|FUWUswWCe}JrJqcCbc||\\wZ?~Z?RnmMwBRp}^IJJ?EuyRjGN]?EWR]eWzp?AJzRefEV??LU^tgdKw?@H\\ai~qM??Ib~Lynns??~hdvzYDg??vm{A~E@O??TqXVHY[w??A|dq^Iio???gjaby{qW??BrX\\C]fV???EI\\tEawn???AAvk|iv_???AZ}wLVOa_???wPW}A~z????F\\JJSMYT????FvKS}^~W????JIvdzw`w????zXUNqYF?????VJbu@l^K????B]|mPHl|?????{xZ\\Kb|g????A\\uscQZF_?????XhFx|YU?????FZtZqIHa??????rQMg~NV??????U{b\\eWNG?????Fo{mBHmr??????AzdRjLTW??????PzpAxR~???????SyFkhypW??????^[d~nIt{???????vkNBVmk???????morxNCng???????gfo]utk_???????',
8728
'DenBip46': '~?@[????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????Ff|JcQhNqs^iBpm{rW~jsQEGVX[[{hCkFBv|lkM]@f^QgrVv?kHTnFWB{Ah}lNumh_ECoEl~E\\_@v\\ScDLz?AtA^nELL??dnFwnfUs?AFw|sjmO??^sErIH[k??o}Vmwdrc??q^WDhSV[??IQEF{}fS???}qoI}ez???RWXx[}T_???uVTxwlGO??F~_sUtjw???BoNHnSVQ_??AmEX]Dz]o???^_ai{vlk???F}Zo@\\VH_???EG|\\}rB{????v|GPVlgG????LI|Q{Jbc????J^TlIAnC????FFJLYzgc?????KZjnn`K??????w^J^hwaO?????Rf]@U\\m_????EL~AJVFW_????@HCFt^]r_?????rJVkaO[{?????CIplZnm}??????M_]VIlz_??????anCKaN~S??????r?teX]es??????@bwgv}Ay??????FnrtN\\S@_??????}{u\\^K{[??????@AY[Y^[j_??????DJ}}rmcZ???????Fwo}KTv`_???????',
8729
'DenBip47': '~?@]????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????FmQbhp{qWV@jyD~ueR`rsUfUT[~{[fHq@xPIzhfF_vdMn{miG@QrItJ[jgCiBg{^ZboDtmCW}}io?Yk~ong`g?Q`Ppnf^S?EIRi~sAlO?Z}GSd}Lw??hQY]Dxui??NZm{Vad_??Tf]RdsD[??BuV~Mhcro??CjuZnw]m??B@[mYdvFo???cg]~scfo??@}bwcJVr???BSTx_Z[n_???}oay[]ZE???BskrnaKdO???HzuFRY[c????wVPvmym?????ZXVUzh@Y????ZvamlBZl????FUtdQdPl_????~uaMXLQK????BmXvPdMxO????@Bedxiimo????BkgZVRb[O????BCFn@^i}o?????Umz`iWtU?????DnQGmmm{??????MdzSeVso??????j[sR}D{U??????mWYKR~rm??????ItrE^Y{C??????CJkoYX{yo??????JqJ\\eZ[K??????BA~ncSX`W??????CJ^fs^YuO??????FVnz?BY\\o???????tpt]sClW???????u~g{qRnm????????',
8730
'DenBip48': '~?@_????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????Im^xtIcWkPV}o}ooP[sZpINJepbxTtyW_IB|_JzDn?GF]ymRrKA^D}{i]kOAFx^G@flgB|j||Z_]c?|mf?Ysy`?CxrmpJPJW?AqmyJ}Xo??H^yBasxT??YBMttfTo??OUaev\\jR??EVVg`bKZo??vHsITX[]??@}^EkQzes??FnlVzSGzo??A]q}biIeO???b]dz}ArK???nmuVMViK???@t[\\N|]{g???U[pHuM}c_???tPHvTF]W????qH|Khm\\z????IwCTzqn^????A|vHnwsWw????}XWlQ^yu????BzLYgden_????Dy}GQ~FQ?????FCZ~_VKlg????Bt~RV`y\\g?????HmIIzSnU?????FCTnH^l?g?????HJRG|HB~_?????|in{_KWk??????JMzQZJV@??????S^H\\LRQf_?????BhgP~lP^w???????\\QxK~uF??????@^YBRQTlc??????C\\ef{Hv@_??????FmHh_MleW??????A~LsZmLH{???????{xE`tTBy????????A`bu|Zfg???????AyRC~raR_???????',
8731
'DenBip49': '~?@a????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????id{X}]FRGzdQpT}teWJZbwwp{dw~fiKKQ`Qyl\\az]CFcpfQDT^c@hoxHbZ|c@}vbbvl_k?D~cku{LN?B~Kqvmr_g?If]azRXCo?]X^JXtue??rdK~exUO??JOhph~YJ_?C]ztbBH~c??_u}Oknpt??Blk|iJWYS???yfFV_jiw??EChpiXfxw??@@qQ~lUvC???lpjf`{gI_??FMxpv~\\[????QLqZol{JO???]LtSBLTv_???}WxTL_yi_???]^bvKCY^_???CWq~[Wvx?????[i|wxTHm_???ATvtduV?Q?????x^||~^HG????FSb[rtLIc????BMFMpUSzk?????VQ|?p^VQ_????AFsEEzqV{?????EPx@drxmo?????Nyfw~UJI??????Jx@vFbKx_?????DFmMhygfO?????FXAyJct|{??????xJfhfUb|_?????APyLwJknE??????Dly_ZrDE[??????FRyN\\oBdc???????oDnkx~}q???????z{c^Dhaw???????FUwvckC]c???????DxYET[jl_???????lyXuuo]r????????~SIFxzou_????????',
8732
'DenBip50': '~?@c????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????QveUtG}auj_t`{bQ}NT`R^fh?rV@s]OC||FKgYFHtLyB~utLO{Vi?F^~KF{[~?yrQql{HEO@l]E}}_Ec?WIdnXk~Ag?zkjIZ@yy_?PK`Xf{vMO?TJZvHPNQw?BdRFnlV]_??Q|y[XWbl??BXyjcel^Y??DdTUQbqVM??ENy}rQh@A??BWAm~_Vup???VHNm[bw|O??FGzuZscXG????Li@^|~]G???ZnhZaWTq_???iq[f?rXz_???JNjQmhWLO????L{[h{kpM????]pWzPrFB_????oUVxuX~H????C\\Tg|R{`Q????EWnhYPGxu????@yvnSFbw[?????nIs\\P}CBo????D^Z_K~reM?????AMRkP~Kn_?????fRVBu{cxo?????c}HukeGto?????Hhd^XYj}_?????ButRaxX\\???????ftC`|]yf_?????@^A|M`jMq??????DwcnAf~][??????@Jv[kwc{K??????Aa?Vqxv^K???????~kUh[d?ZO??????@P^XVXJPq???????]cI@~dy}G???????spxV}FL~????????LI~L_x^Z_???????LW]OQZVnW???????EiLlk|S_[????????',
8733
'K02': 'A_',
8734
'K02K03': 'E{Sw',
8735
'K02K04': 'G~`HW{',
8736
'K02K05': 'I~}AHKVBw',
8737
'K02K06': 'K~~{ACbCwV_~',
8738
'K02K07': 'M~~~{@@GWb`N@^?~_',
8739
'K02K08': 'O~~~~}?OH@aFAN@N_VwB~',
8740
'K02K09': 'Q~~~~~~_A?cBCFAF_bwC~?V{?~w',
8741
'K02K10': 'S~~~~~~~{?G@GBCB`@wG^?b{@Nw@^w?~{',
8742
'K02K11': 'U~~~~~~~~~o?O@G@a?wON@@{AFwANw@N{?V~?B~w',
8743
'K02K12': 'W~~~~~~~~~~~_?O?c?W_FA?{CBwCFwAF{?b~?C~w?V~_?~~',
8744
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8793
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8794
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8795
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8796
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8797
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8798
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8799
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8800
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8801
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8802
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8803
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8804
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8805
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8806
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8807
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8808
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8809
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8810
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8811
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8812
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8813
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8814
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8815
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8816
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8817
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8818
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8819
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8820
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8821
'K05K15': '~?@J~~~~~~~~~~~~~~~~~{??O?C_?W_?wO?{C?^?_FwA?~_CB~?CF~?AF~_?b~w?C~~??V~{??~~{??_?A??O?C_?C?BC??_?wO?A?F__?C?^?_?C?~?O?A?~_C??_^w?_?CF~?A??O~{?C??b~w?C??f~w?A??V~{??_?F~~_?C??_?A??O?A??c??_?C?BC??_?C?FA??O?A?F__?C??_BwC??_?C?~?O?A??OF{?_?C??_^w?_?C??_~w?O?A??O~{?C??_?C^~??_?C??f~w?A??O?A~~_?C??_?F~~_?C??_?C??O?A??O?A??c??_?C??_?W_?C??_?C?FA??O?A??O?{C??_?C??_BwC??_?C??_FwA??O?A??OF{?_?C??_?CB~?C??_?C??_~w?O?A??O?AF~_?_?C??_?C^~??_?C??_?C~~??O?A??O?A~~_?C??_?C??~~w',
8822
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8823
'K05K17': '~?@T~~~~~~~~~~~~~~~~~~~~~~}??A??H??KO?FA?@wG?N_O?~?O@~?G@~_A?~w?ON~?@@~{?AF~w?AN~w?@N~{??V~~??B~~{??G??G??O?@G??O?BC??G?B`??A?@wG??O?^?_?@?B{@??A?Nw@??A?^w?_?@?^{?G??ON~?@??AB~w?C??G^~_?G??P~~??G??R~~??C??J~~_?@??B~~{??G??O??O??_?@??C_?@??A??W_?@??A??wO??_?@??{C??G??O?^?_?@??A?FwA??C??G?~_C??G??OB~?C??G??OF~?A??C??GF~_?_?@??AB~w?C??G??O~~??O??_?@F~{??_?@??A^~w??_?@??A~~w??O??_?@~~}??C??G??O??O??_?@??A??H??A??C??G?@a??C??G??O?FA??C??G??O?N@??A??C??G?N_O??_?@??A?FwA??C??G??O@~?G??O??_?@?N{?O??_?@??A?~w?O??_?@??A@~w?G??O??_?@@~{?A??C??G??O~~??O??_?@??AN~w?@??A??C??H~~_?A??C??G??V~~??A??C??G??^~~',
8824
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8825
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8826
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8827
'K06': 'E~~w',
8828
'K06K06': 'c~~{ACbCwV_~__OOcCW_fAA{CF{CCAAAC__bCCCwOOV___~____OOOOcCCCW___fAAAA{CCCF{CCCCAAAAAC____bCCCCCwOOOOV_____~',
8829
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8830
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8831
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8832
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8833
'K06K11': '~?@A~~~~~~~~~o?O@G@a?wON@@{AFwANw@N{?V~?B~{?G?G?O@G?OBC?GB`?A@wG?O^?_@B{@?ANw@?A^w?_@^{?G?^~_@?A?A?C?G?c?G?OBC?G?OFA?C?GF__@?ABwC?G?O~?O?_@F{?_@?A^w?_@?A~w?O?_@~}?C?G?O?O?_@?A?H?A?C?G@a?C?G?OFA?C?G?ON@?A?C?GN_O?_@?AFwA?C?G?P~?G?O?_@N{?O?_@?A~w?O?_@?B~{?G?O?_@?@?A?C?G?O@G?O?_@?A?W_@?A?C?GB`?A?C?G?ON@?A?C?G?O^?_@?A?C?G^_G?O?_@?ANw@?A?C?G?R~?C?G?O?_@^{?G?O?_@?B~w',
8834
'K06K12': '~?@G~~~~~~~~~~~_?O?c?W_FA?{CBwCFwAF{?b~?C~w?V~_?~~_?_?O?O?c?C?W_?_FA?A?{C?CBwC?CFwA?AF{?_?b~?C?C~w?O?V~_?_?~~_?_?_?O?O?O?c?C?C?W_?_?_FA?A?A?{C?C?CBwC?C?CFwA?A?AF{?_?_?b~?C?C?C~w?O?O?V~_?_?_?~~_?_?_?_?O?O?O?O?c?C?C?C?W_?_?_?_FA?A?A?A?{C?C?C?CBwC?C?C?CFwA?A?A?AF{?_?_?_?b~?C?C?C?C~w?O?O?O?V~_?_?_?_?~~_?_?_?_?_?O?O?O?O?O?c?C?C?C?C?W_?_?_?_?_FA?A?A?A?A?{C?C?C?C?CBwC?C?C?C?CFwA?A?A?A?AF{?_?_?_?_?b~?C?C?C?C?C~w?O?O?O?O?V~_?_?_?_?_?~~',
8835
'K06K13': '~?@M~~~~~~~~~~~~~_?G?H?BC?[G@wGBwCB{@@~?G^w?b~_@N~?@^~??~~o?G?A?@?@G?C?KO?G?wO?G@wG?C@{A?@?~?O?GNw@??`~_A?@F~?A?@N~?@??n~_?O?N~{?A?@??O?G?C?C_?O?G?W_?O?G?wO?G?C?{C?A?@?^?_?O?GFwA?@??_~_C?A?@B~?C?A?@F~?A?@??f~_?_?O?J~w?C?A?@~~_?O?G?C?@??_?O?G?H??_?O?G?W_?O?G?C?[G?C?A?@?N@??_?O?GBwC?A?@??_^_G?C?A?@@~?G?C?A?@B~?C?A?@??b~_@??_?O?H~w?G?C?A?@^~??_?O?G?F~}?@??_?O?G?A?@??_?O?G?H??_?O?G?C?KO?G?C?A?@?FA?@??_?O?G@wG?C?A?@??_N_O?G?C?A?@?~?O?G?C?A?@@~?G?C?A?@??`~_A?@??_?O?G~w?O?G?C?A?@N~?@??_?O?G?D~{?A?@??_?O?N~w',
8836
'K06K14': '~?@S~~~~~~~~~~~~~~~o?A?@G?KO?wO@wG@{A?~?ONw@@~_AF~?AN~?@N~_?V~w?B~~_?G?@??O?H??O?W_?G?[G?A?N@??OBwC?@?^_G?A@~?G?AB~?C?@B~_@??P~w?G?A^~??_?J~{?@??^~{?@??O?A??_?G?C_?G?A?BC?@??O?wO?C?@?F__?G?A?^?_?G?A?~?O?C?@?~_C?@??O^w?_?G?AF~?A??_?G~{?C?@??R~w?C?@??V~w?A??_?N~}??_?G?A??O?C?@??O?H??O?C?@?@a??_?G?A?FA??_?G?A?N@??O?C?@?N_O?C?@??OFwA??_?G?A@~?G?A??_?GN{?O?C?@??O~w?O?C?@??P~w?G?A??_?H~{?A??_?G?A~~??O?C?@??^~{?@??O?C?@??G?A??_?G?A?@G?A??_?G?A?BC?@??O?C?@?B`??O?C?@??O@wG?A??_?G?A?^?_?G?A??_?GB{@??O?C?@??ONw@??O?C?@??O^w?_?G?A??_?G^{?G?A??_?G?AN~?@??O?C?@??R~w?C?@??O?C?@^~_?G?A??_?G?B~~',
8837
'K06K15': '~?@Y~~~~~~~~~~~~~~~~~{??O?C_?W_?wO?{C?^?_FwA?~_CB~?CF~?AF~_?b~w?C~~??V~{??~~{??_?A??O?C_?C?BC??_?wO?A?F__?C?^?_?C?~?O?A?~_C??_^w?_?CF~?A??O~{?C??b~w?C??f~w?A??V~{??_?F~~_?C??_?A??O?A??c??_?C?BC??_?C?FA??O?A?F__?C??_BwC??_?C?~?O?A??OF{?_?C??_^w?_?C??_~w?O?A??O~{?C??_?C^~??_?C??f~w?A??O?A~~_?C??_?F~~_?C??_?C??O?A??O?A??c??_?C??_?W_?C??_?C?FA??O?A??O?{C??_?C??_BwC??_?C??_FwA??O?A??OF{?_?C??_?CB~?C??_?C??_~w?O?A??O?AF~_?_?C??_?C^~??_?C??_?C~~??O?A??O?A~~_?C??_?C??~~{??_?C??_?C??O?A??O?A??O?C_?C??_?C??_?W_?C??_?C??_?wO?A??O?A??O?{C??_?C??_?C?^?_?C??_?C??_FwA??O?A??O?A?~_C??_?C??_?CB~?C??_?C??_?CF~?A??O?A??O?AF~_?_?C??_?C??b~w?C??_?C??_?C~~??O?A??O?A??V~{??_?C??_?C??~~w',
8838
'K06K16': '~?@_~~~~~~~~~~~~~~~~~~~~_?@??H??W_?[G?N@?BwC?^_G@~?GB~?CB~_@@~w?G^~??b~{?@N~w?@^~w??~~}??G??O?@??H??C?@a??G?FA??G?N@??C?N_O?@?FwA??G@~?G??_N{?O?@?~w?O?@@~w?G??`~{?A??G~~??O?@N~w?@??D~~_?A??N~~_?A??G??O?@??C??c??O?@??W_?A??G?FA??G??_?{C??O?@?BwC??O?@?FwA??G??_F{?_?A??GB~?C??O?@?~w?O?@??CF~_?_?A??G^~??_?A??G~~??O?@??C~~_?C??O?@^~w??_?A??N~~_?A??G??_?@??C??O?@??H??C??O?@??W_?A??G??_?[G??_?A??G?N@??C??O?@?BwC??O?@??C?^_G??_?A??G@~?G??_?A??GB~?C??O?@??CB~_@??C??O?@@~w?G??_?A??G^~??_?A??G??b~{?@??C??O?@N~w?@??C??O?@^~w??_?A??G??~~}??G??_?A??G??O?@??C??O?@??H??C??O?@??C?@a??G??_?A??G?FA??G??_?A??G?N@??C??O?@??C?N_O?@??C??O?@?FwA??G??_?A??G@~?G??_?A??G??_N{?O?@??C??O?@?~w?O?@??C??O?@@~w?G??_?A??G??`~{?A??G??_?A??G~~??O?@??C??O?@N~w?@??C??O?@??D~~_?A??G??_?A??N~~',
8839
'K07': 'F~~~w',
8840
'K07K07': 'p~~~{@@GWb`N@^?~oGA@@GCKOGwOHwGD{A@~_OGA@?_cA@BCA@FA@?f__OJwCA@~_OGC@?_OGH?_OGW_OGC[GCA@N@?_OJwCA@?~oGCA@?OGCA@@GCA@?`a@?_OGwOGCA@N@?_OGD{A@?_ON{A@?_OGA@?_OGCC_OGCA@BCA@?_OGwOGCA@?f__OGCA@^?_OGCA@~',
8841
'K07K08': 'w~~~~}?OH@aFAN@N_VwB~_G@?OH?OW_G[GAN@?RwC@^_GB~_GA?OC@?c@?OW_GAFA?_G{C@?RwC@?VwA?_N}?_GA?OC@?OH?OC@@a?_GAFA?_GAN@?OC@N_OC@?VwA?_GB~_GA?_G@?OC@?OH?OC@?OW_GA?_G[GA?_GAN@?OC@?RwC@?OC@^_GA?_GB~_GA?_GA?OC@?OC@?c@?OC@?OW_GA?_GAFA?_GA?_G{C@?OC@?RwC@?OC@?VwA?_GA?_N{',
8842
'K07K09': '~??~~~~~~~_A?cBCFAF_bwC~?V{?~{?_A?OC_CBC?_wOAF__C^?_C~?OA~_C?~{?_C?OA?OC_C?_W_C?_wOA?O{C?_C^?_C?fwA?OA~_C?_F~_C?_C?OA?OA?c?_C?_W_C?_CFA?OA?O{C?_C?bwC?_C?fwA?OA?V{?_C?_F~_C?_C?_A?OA?OA?c?_C?_CBC?_C?_CFA?OA?OAF__C?_C?bwC?_C?_C~?OA?OA?V{?_C?_C?~{?_C?_C?_A?OA?OA?OC_C?_C?_CBC?_C?_C?_wOA?OA?OAF__C?_C?_C^?_C?_C?_C~?OA?OA?OA~_C?_C?_C?~w',
8843
'K07K10': '~?@E~~~~~~~{?G@GBCB`@wG^?b{@Nw@^w?~}?G?O@?H?C@a?GFA?GN@?CN_O@FwA?H~?G?n{?O@~{?O@?A?G?_C_A?GBC?O@?wO@?CF__A?G^?_A?G~?O@?C~_C?O@^w?_A?N~_A?G?_@?C?O@?H?C?O@?W_A?G?_[G?_A?GN@?C?O@BwC?O@?C^_G?_A?H~?G?_A?J~?C?O@?F~o@?C?O@?A?G?_A?G@G?_A?G?_KO@?C?O@?wO@?C?O@@wG?_A?G?`{A?G?_A?G~?O@?C?O@Nw@?C?O@?D~_A?G?_A?N~_A?G?_A?G?O@?C?O@?C?c?O@?C?O@?W_A?G?_A?GFA?G?_A?G?_{C?O@?C?O@BwC?O@?C?O@FwA?G?_A?G?f{?_A?G?_A?J~?C?O@?C?O@~w',
8844
'K07K11': '~?@L~~~~~~~~~o?O@G@a?wON@@{AFwANw@N{?V~?B~{?G?G?O@G?OBC?GB`?A@wG?O^?_@B{@?ANw@?A^w?_@^{?G?^~_@?A?A?C?G?c?G?OBC?G?OFA?C?GF__@?ABwC?G?O~?O?_@F{?_@?A^w?_@?A~w?O?_@~}?C?G?O?O?_@?A?H?A?C?G@a?C?G?OFA?C?G?ON@?A?C?GN_O?_@?AFwA?C?G?P~?G?O?_@N{?O?_@?A~w?O?_@?B~{?G?O?_@?@?A?C?G?O@G?O?_@?A?W_@?A?C?GB`?A?C?G?ON@?A?C?G?O^?_@?A?C?G^_G?O?_@?ANw@?A?C?G?R~?C?G?O?_@^{?G?O?_@?B~{?G?O?_@?A?A?C?G?O?_@?C_@?A?C?G?OBC?G?O?_@?A?wO?_@?A?C?GF__@?A?C?G?O^?_@?A?C?G?O~?O?_@?A?C?G~_C?G?O?_@?A^w?_@?A?C?G?V~?A?C?G?O?_@~{',
8845
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8846
'K07K13': '~?@Z~~~~~~~~~~~~~_?G?H?BC?[G@wGBwCB{@@~?G^w?b~_@N~?@^~??~~o?G?A?@?@G?C?KO?G?wO?G@wG?C@{A?@?~?O?GNw@??`~_A?@F~?A?@N~?@??n~_?O?N~{?A?@??O?G?C?C_?O?G?W_?O?G?wO?G?C?{C?A?@?^?_?O?GFwA?@??_~_C?A?@B~?C?A?@F~?A?@??f~_?_?O?J~w?C?A?@~~_?O?G?C?@??_?O?G?H??_?O?G?W_?O?G?C?[G?C?A?@?N@??_?O?GBwC?A?@??_^_G?C?A?@@~?G?C?A?@B~?C?A?@??b~_@??_?O?H~w?G?C?A?@^~??_?O?G?F~}?@??_?O?G?A?@??_?O?G?H??_?O?G?C?KO?G?C?A?@?FA?@??_?O?G@wG?C?A?@??_N_O?G?C?A?@?~?O?G?C?A?@@~?G?C?A?@??`~_A?@??_?O?G~w?O?G?C?A?@N~?@??_?O?G?D~{?A?@??_?O?N~{?A?@??_?O?G?A?@??_?O?G?C?C_?O?G?C?A?@?BC?A?@??_?O?G?wO?G?C?A?@??_F__?O?G?C?A?@?^?_?O?G?C?A?@?~?O?G?C?A?@??_~_C?A?@??_?O?G^w?_?O?G?C?A?@F~?A?@??_?O?G?C~{?C?A?@??_?O?J~w?C?A?@??_?O?N~w',
8847
'K07K14': '~?@a~~~~~~~~~~~~~~~o?A?@G?KO?wO@wG@{A?~?ONw@@~_AF~?AN~?@N~_?V~w?B~~_?G?@??O?H??O?W_?G?[G?A?N@??OBwC?@?^_G?A@~?G?AB~?C?@B~_@??P~w?G?A^~??_?J~{?@??^~{?@??O?A??_?G?C_?G?A?BC?@??O?wO?C?@?F__?G?A?^?_?G?A?~?O?C?@?~_C?@??O^w?_?G?AF~?A??_?G~{?C?@??R~w?C?@??V~w?A??_?N~}??_?G?A??O?C?@??O?H??O?C?@?@a??_?G?A?FA??_?G?A?N@??O?C?@?N_O?C?@??OFwA??_?G?A@~?G?A??_?GN{?O?C?@??O~w?O?C?@??P~w?G?A??_?H~{?A??_?G?A~~??O?C?@??^~{?@??O?C?@??G?A??_?G?A?@G?A??_?G?A?BC?@??O?C?@?B`??O?C?@??O@wG?A??_?G?A?^?_?G?A??_?GB{@??O?C?@??ONw@??O?C?@??O^w?_?G?A??_?G^{?G?A??_?G?AN~?@??O?C?@??R~w?C?@??O?C?@^~_?G?A??_?G?B~~_?G?A??_?G?A??O?C?@??O?C?@??c?@??O?C?@??O?W_?G?A??_?G?A?FA??_?G?A??_?G?{C?@??O?C?@??OBwC?@??O?C?@??OFwA??_?G?A??_?GF{?_?G?A??_?G?AB~?C?@??O?C?@??O~w?O?C?@??O?C?@F~_?_?G?A??_?G?A^~??_?G?A??_?G?A~~??O?C?@??O?C?@~~_',
8848
'K08': 'G~~~~{',
8849
'K08K08': '~?@?~~~~}?OH@aFAN@N_VwB~_G@?OH?OW_G[GAN@?RwC@^_GB~_GA?OC@?c@?OW_GAFA?_G{C@?RwC@?VwA?_N}?_GA?OC@?OH?OC@@a?_GAFA?_GAN@?OC@N_OC@?VwA?_GB~_GA?_G@?OC@?OH?OC@?OW_GA?_G[GA?_GAN@?OC@?RwC@?OC@^_GA?_GB~_GA?_GA?OC@?OC@?c@?OC@?OW_GA?_GAFA?_GA?_G{C@?OC@?RwC@?OC@?VwA?_GA?_N}?_GA?_GA?OC@?OC@?OH?OC@?OC@@a?_GA?_GAFA?_GA?_GAN@?OC@?OC@N_OC@?OC@?VwA?_GA?_GB~',
8850
'K08K09': '~?@G~~~~~~_A?cBCFAF_bwC~?V{?~{?_A?OC_CBC?_wOAF__C^?_C~?OA~_C?~{?_C?OA?OC_C?_W_C?_wOA?O{C?_C^?_C?fwA?OA~_C?_F~_C?_C?OA?OA?c?_C?_W_C?_CFA?OA?O{C?_C?bwC?_C?fwA?OA?V{?_C?_F~_C?_C?_A?OA?OA?c?_C?_CBC?_C?_CFA?OA?OAF__C?_C?bwC?_C?_C~?OA?OA?V{?_C?_C?~{?_C?_C?_A?OA?OA?OC_C?_C?_CBC?_C?_C?_wOA?OA?OAF__C?_C?_C^?_C?_C?_C~?OA?OA?OA~_C?_C?_C?~{?_C?_C?_C?OA?OA?OA?OC_C?_C?_C?_W_C?_C?_C?_wOA?OA?OA?O{C?_C?_C?_C^?_C?_C?_C?fwA?OA?OA?OA~_C?_C?_C?_F~',
8851
'K08K10': '~?@O~~~~~~~{?G@GBCB`@wG^?b{@Nw@^w?~}?G?O@?H?C@a?GFA?GN@?CN_O@FwA?H~?G?n{?O@~{?O@?A?G?_C_A?GBC?O@?wO@?CF__A?G^?_A?G~?O@?C~_C?O@^w?_A?N~_A?G?_@?C?O@?H?C?O@?W_A?G?_[G?_A?GN@?C?O@BwC?O@?C^_G?_A?H~?G?_A?J~?C?O@?F~o@?C?O@?A?G?_A?G@G?_A?G?_KO@?C?O@?wO@?C?O@@wG?_A?G?`{A?G?_A?G~?O@?C?O@Nw@?C?O@?D~_A?G?_A?N~_A?G?_A?G?O@?C?O@?C?c?O@?C?O@?W_A?G?_A?GFA?G?_A?G?_{C?O@?C?O@BwC?O@?C?O@FwA?G?_A?G?f{?_A?G?_A?J~?C?O@?C?O@~{?O@?C?O@?C?G?_A?G?_A?G@G?_A?G?_A?GBC?O@?C?O@?CB`?C?O@?C?O@@wG?_A?G?_A?G^?_A?G?_A?G?b{@?C?O@?C?O@Nw@?C?O@?C?O@^w?_A?G?_A?G?~{',
8852
'K08K11': '~?@W~~~~~~~~~o?O@G@a?wON@@{AFwANw@N{?V~?B~{?G?G?O@G?OBC?GB`?A@wG?O^?_@B{@?ANw@?A^w?_@^{?G?^~_@?A?A?C?G?c?G?OBC?G?OFA?C?GF__@?ABwC?G?O~?O?_@F{?_@?A^w?_@?A~w?O?_@~}?C?G?O?O?_@?A?H?A?C?G@a?C?G?OFA?C?G?ON@?A?C?GN_O?_@?AFwA?C?G?P~?G?O?_@N{?O?_@?A~w?O?_@?B~{?G?O?_@?@?A?C?G?O@G?O?_@?A?W_@?A?C?GB`?A?C?G?ON@?A?C?G?O^?_@?A?C?G^_G?O?_@?ANw@?A?C?G?R~?C?G?O?_@^{?G?O?_@?B~{?G?O?_@?A?A?C?G?O?_@?C_@?A?C?G?OBC?G?O?_@?A?wO?_@?A?C?GF__@?A?C?G?O^?_@?A?C?G?O~?O?_@?A?C?G~_C?G?O?_@?A^w?_@?A?C?G?V~?A?C?G?O?_@~}?C?G?O?_@?A?A?C?G?O?_@?A?H?A?C?G?O?_@?KO?_@?A?C?G?OFA?C?G?O?_@?A@wG?O?_@?A?C?GN_O?_@?A?C?G?O~?O?_@?A?C?G?P~?G?O?_@?A?C?H~_A?C?G?O?_@?A~w?O?_@?A?C?G?^~',
8853
'K08K12': '~?@_~~~~~~~~~~~_?O?c?W_FA?{CBwCFwAF{?b~?C~w?V~_?~~_?_?O?O?c?C?W_?_FA?A?{C?CBwC?CFwA?AF{?_?b~?C?C~w?O?V~_?_?~~_?_?_?O?O?O?c?C?C?W_?_?_FA?A?A?{C?C?CBwC?C?CFwA?A?AF{?_?_?b~?C?C?C~w?O?O?V~_?_?_?~~_?_?_?_?O?O?O?O?c?C?C?C?W_?_?_?_FA?A?A?A?{C?C?C?CBwC?C?C?CFwA?A?A?AF{?_?_?_?b~?C?C?C?C~w?O?O?O?V~_?_?_?_?~~_?_?_?_?_?O?O?O?O?O?c?C?C?C?C?W_?_?_?_?_FA?A?A?A?A?{C?C?C?C?CBwC?C?C?C?CFwA?A?A?A?AF{?_?_?_?_?b~?C?C?C?C?C~w?O?O?O?O?V~_?_?_?_?_?~~_?_?_?_?_?_?O?O?O?O?O?O?c?C?C?C?C?C?W_?_?_?_?_?_FA?A?A?A?A?A?{C?C?C?C?C?CBwC?C?C?C?C?CFwA?A?A?A?A?AF{?_?_?_?_?_?b~?C?C?C?C?C?C~w?O?O?O?O?O?V~_?_?_?_?_?_?~~_?_?_?_?_?_?_?O?O?O?O?O?O?O?c?C?C?C?C?C?C?W_?_?_?_?_?_?_FA?A?A?A?A?A?A?{C?C?C?C?C?C?CBwC?C?C?C?C?C?CFwA?A?A?A?A?A?AF{?_?_?_?_?_?_?b~?C?C?C?C?C?C?C~w?O?O?O?O?O?O?V~_?_?_?_?_?_?_?~~',
8854
'K09': 'H~~~~~~',
8855
'K09K09': '~?@P~~~~~~_A?cBCFAF_bwC~?V{?~{?_A?OC_CBC?_wOAF__C^?_C~?OA~_C?~{?_C?OA?OC_C?_W_C?_wOA?O{C?_C^?_C?fwA?OA~_C?_F~_C?_C?OA?OA?c?_C?_W_C?_CFA?OA?O{C?_C?bwC?_C?fwA?OA?V{?_C?_F~_C?_C?_A?OA?OA?c?_C?_CBC?_C?_CFA?OA?OAF__C?_C?bwC?_C?_C~?OA?OA?V{?_C?_C?~{?_C?_C?_A?OA?OA?OC_C?_C?_CBC?_C?_C?_wOA?OA?OAF__C?_C?_C^?_C?_C?_C~?OA?OA?OA~_C?_C?_C?~{?_C?_C?_C?OA?OA?OA?OC_C?_C?_C?_W_C?_C?_C?_wOA?OA?OA?O{C?_C?_C?_C^?_C?_C?_C?fwA?OA?OA?OA~_C?_C?_C?_F~_C?_C?_C?_C?OA?OA?OA?OA?c?_C?_C?_C?_W_C?_C?_C?_CFA?OA?OA?OA?O{C?_C?_C?_C?bwC?_C?_C?_C?fwA?OA?OA?OA?V{?_C?_C?_C?_F~',
8856
'K09K10': '~?@Y~~~~~~~{?G@GBCB`@wG^?b{@Nw@^w?~}?G?O@?H?C@a?GFA?GN@?CN_O@FwA?H~?G?n{?O@~{?O@?A?G?_C_A?GBC?O@?wO@?CF__A?G^?_A?G~?O@?C~_C?O@^w?_A?N~_A?G?_@?C?O@?H?C?O@?W_A?G?_[G?_A?GN@?C?O@BwC?O@?C^_G?_A?H~?G?_A?J~?C?O@?F~o@?C?O@?A?G?_A?G@G?_A?G?_KO@?C?O@?wO@?C?O@@wG?_A?G?`{A?G?_A?G~?O@?C?O@Nw@?C?O@?D~_A?G?_A?N~_A?G?_A?G?O@?C?O@?C?c?O@?C?O@?W_A?G?_A?GFA?G?_A?G?_{C?O@?C?O@BwC?O@?C?O@FwA?G?_A?G?f{?_A?G?_A?J~?C?O@?C?O@~{?O@?C?O@?C?G?_A?G?_A?G@G?_A?G?_A?GBC?O@?C?O@?CB`?C?O@?C?O@@wG?_A?G?_A?G^?_A?G?_A?G?b{@?C?O@?C?O@Nw@?C?O@?C?O@^w?_A?G?_A?G?~}?G?_A?G?_A?G?O@?C?O@?C?O@?H?C?O@?C?O@?C@a?G?_A?G?_A?GFA?G?_A?G?_A?GN@?C?O@?C?O@?CN_O@?C?O@?C?O@FwA?G?_A?G?_A?H~?G?_A?G?_A?G?n{?O@?C?O@?C?O@~w',
8857
'K09K11': '~?@b~~~~~~~~~o?O@G@a?wON@@{AFwANw@N{?V~?B~{?G?G?O@G?OBC?GB`?A@wG?O^?_@B{@?ANw@?A^w?_@^{?G?^~_@?A?A?C?G?c?G?OBC?G?OFA?C?GF__@?ABwC?G?O~?O?_@F{?_@?A^w?_@?A~w?O?_@~}?C?G?O?O?_@?A?H?A?C?G@a?C?G?OFA?C?G?ON@?A?C?GN_O?_@?AFwA?C?G?P~?G?O?_@N{?O?_@?A~w?O?_@?B~{?G?O?_@?@?A?C?G?O@G?O?_@?A?W_@?A?C?GB`?A?C?G?ON@?A?C?G?O^?_@?A?C?G^_G?O?_@?ANw@?A?C?G?R~?C?G?O?_@^{?G?O?_@?B~{?G?O?_@?A?A?C?G?O?_@?C_@?A?C?G?OBC?G?O?_@?A?wO?_@?A?C?GF__@?A?C?G?O^?_@?A?C?G?O~?O?_@?A?C?G~_C?G?O?_@?A^w?_@?A?C?G?V~?A?C?G?O?_@~}?C?G?O?_@?A?A?C?G?O?_@?A?H?A?C?G?O?_@?KO?_@?A?C?G?OFA?C?G?O?_@?A@wG?O?_@?A?C?GN_O?_@?A?C?G?O~?O?_@?A?C?G?P~?G?O?_@?A?C?H~_A?C?G?O?_@?A~w?O?_@?A?C?G?^~_@?A?C?G?O?_@?@?A?C?G?O?_@?A?H?A?C?G?O?_@?A?W_@?A?C?G?O?_@?[G?O?_@?A?C?G?ON@?A?C?G?O?_@?ABwC?G?O?_@?A?C?G^_G?O?_@?A?C?G?P~?G?O?_@?A?C?G?R~?C?G?O?_@?A?C?J~_@?A?C?G?O?_@?B~w',
8858
'K10': 'I~~~~~~~w',
8859
'K10K10': '~?@c~~~~~~~{?G@GBCB`@wG^?b{@Nw@^w?~}?G?O@?H?C@a?GFA?GN@?CN_O@FwA?H~?G?n{?O@~{?O@?A?G?_C_A?GBC?O@?wO@?CF__A?G^?_A?G~?O@?C~_C?O@^w?_A?N~_A?G?_@?C?O@?H?C?O@?W_A?G?_[G?_A?GN@?C?O@BwC?O@?C^_G?_A?H~?G?_A?J~?C?O@?F~o@?C?O@?A?G?_A?G@G?_A?G?_KO@?C?O@?wO@?C?O@@wG?_A?G?`{A?G?_A?G~?O@?C?O@Nw@?C?O@?D~_A?G?_A?N~_A?G?_A?G?O@?C?O@?C?c?O@?C?O@?W_A?G?_A?GFA?G?_A?G?_{C?O@?C?O@BwC?O@?C?O@FwA?G?_A?G?f{?_A?G?_A?J~?C?O@?C?O@~{?O@?C?O@?C?G?_A?G?_A?G@G?_A?G?_A?GBC?O@?C?O@?CB`?C?O@?C?O@@wG?_A?G?_A?G^?_A?G?_A?G?b{@?C?O@?C?O@Nw@?C?O@?C?O@^w?_A?G?_A?G?~}?G?_A?G?_A?G?O@?C?O@?C?O@?H?C?O@?C?O@?C@a?G?_A?G?_A?GFA?G?_A?G?_A?GN@?C?O@?C?O@?CN_O@?C?O@?C?O@FwA?G?_A?G?_A?H~?G?_A?G?_A?G?n{?O@?C?O@?C?O@~{?O@?C?O@?C?O@?A?G?_A?G?_A?G?_C_A?G?_A?G?_A?GBC?O@?C?O@?C?O@?wO@?C?O@?C?O@?CF__A?G?_A?G?_A?G^?_A?G?_A?G?_A?G~?O@?C?O@?C?O@?C~_C?O@?C?O@?C?O@^w?_A?G?_A?G?_A?N~',
8860
'K11': 'J~~~~~~~~~_',
8861
'K12': 'K~~~~~~~~~~~',
8862
'K13': 'L~~~~~~~~~~~~~',
8863
'K14': 'M~~~~~~~~~~~~~~~_',
8864
'K15': 'N~~~~~~~~~~~~~~~~~w',
8865
'K16': 'O~~~~~~~~~~~~~~~~~~~~',
8866
'K17': 'P~~~~~~~~~~~~~~~~~~~~~~{',
8867
'K18': 'Q~~~~~~~~~~~~~~~~~~~~~~~~~w',
8868
'K19': 'R~~~~~~~~~~~~~~~~~~~~~~~~~~~~w',
8869
'K20': 'S~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{',
8870
'Lem': 'GqKp_{',
8871
'LemK02': 'Or`H?cI@WS?hA_@O_TOAj',
8872
'LemK03': 'W{S{ACbC?O_b?c?Q_CwC_?QC?cWC_?AOC?cB?Cc_?QQ_?cf',
8873
'LemK04': '_~`HW}?OH@aFA?@?_OWAF?G_?PG?PW?G{?a??COG?P@_?aF?G_??CO?_@C?W?G_F??aG_?@CPG?@CPW??aG{',
8874
'LemK05': 'g~}AHKVB{?G@GBCB`@wG??_C@?W@?w?_{?GO?@AG?CH_?GV??G^?AC???`?G?CGB??O_[??`@w?O_???GO?C?AC?B??O_?w?@A?F_?ACGO??ACGP??@ACH_??O`Aw??ACG^',
8875
'LemK06': 'o~~{ACbCwV_~_?O?c?W_FA?{CBwC??A?C?_B?C?w?OF_?_^??__??OO_?CCW??_f??AA{??CFw?CC???AA?C??__B??CC?w??OOF_??__^??__????OO??_?CC??W??__?F??AA??{??CC?Bw??CCCC???AAAAC???___b???CCCCw???OOOV_???___~',
8876
'LemK07': 'w~~~{@@GWb`N@^?~o?A?@G?KO?wO@wG@{A?~?O??@??_A?B?A?F?@?F_?OBw?A?~??GC???OGG??OGW??GC[??A@N???OJw??@?~_?CA????CA?@??A@?@_??_O?w??CA?N???OG@{???_OFw?@?_?????_O??C??GC??B??@?_??w??CA??F_??GC??^???GC??~???CA@?_???@?_OGG???GCA@B????_OGC[???@?_OHw???@?_OJw????_OGF{',
8877
'LemK08': '~?@?~~~~}?OH@aFAN@N_VwB~_?@??H??W_?[G?N@?BwC?^_G@~?G???C??_@??W?G?F??_?{?@?Bw?@?Fw??_F{??GA???@?OG??C@@_??GAF???GAN???C@N_??@?Vw???GB~??A?_????C@??G??C@??W??A?_?[???_G?N???C@?Bw???OC?^_???_G@~??A?_?????@?O???_??OC???W??A?_??F???GA???{???OC??Bw???OC??Fw???GA??F{???A?_GA?????OC@?OG???@?OC@@_???A?_GAF????A?_GAN????@?OC@N_????OC@?Vw????A?_GB~',
8878
'LemK09': '~?@G~~~~~~_A?cBCFAF_bwC~?V{?~{??A??C_?BC??wO?F__?^?_?~?O?~_C?^w?_???A??C?C??W?C??w?A??{??_?^??C?Fw??O?~_??_B~???_C????OA?_??C?_W???_CF???A?O{???C?bw???C?fw???A?V{????_F~???_C?????A?O??_??C?_?B???C?_?F???A?O?F_???_C?Bw???C?_?~????OA?F{????_C?^w??C?_??????A?O???C???_C???B???C?_???w???OA???F_???_C???^????_C???~????OA???~_???C?_??^w????_C?_C?????A?OA?OC????C?_C?_W????C?_C?_w????A?OA?O{?????_C?_C^?????C?_C?fw?????OA?OA~_?????_C?_F~',
8879
'LemK10': '~?@O~~~~~~~{?G@GBCB`@wG^?b{@Nw@^w?~}???O??H??@a??FA??N@??N_O?FwA?@~?G?N{?O?~w?O????G??C?A??B??O??w?@??F_?A??^??A??~??@??~_??O?^w??A?F~???G?_????O@?G???O@?W???G?_[???A?GN????O@Bw???@?C^_???A?H~????A?J~????@?F~_??C?O??????_A??@???A?G??K???C?O??w???C?O?@w???A?G?@{????_A??~????C?O?Nw????O@?@~_????_A?F~???G?_???????C?O????_??@?C????W???G?_???F????_A????{???@?C???Bw???@?C???Fw????_A???F{????G?_??B~????@?C???~w????C?O@?C??????G?_A?G@?????G?_A?GB?????C?O@?CB_????@?C?O@@w?????G?_A?G^??????_A?G?b{?????@?C?O@Nw?????@?C?O@^w??????_A?G?~{',
8880
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8881
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8882
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8883
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8884
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8885
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8886
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8887
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8888
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8889
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8890
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8891
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8892
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8893
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8894
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8895
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8896
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8897
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8898
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8899
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8900
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8901
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8902
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8903
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8904
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8905
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8906
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8907
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8908
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8909
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8910
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8911
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8921
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8922
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8923
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8924
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8925
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8926
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8928
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8930
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8931
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8932
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8933
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8934
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8935
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8936
'Q3K10': '~?@O~~~~~~~{?G@GBCB`@wG^?b{@Nw@^w?~}???O??H??@a??FA??N@??N_O?FwA?@~?G?N{?O?~w?O@???G?_C?A?GB??O@?w?@?CF_?A?G^??A?G~??@?C~_??O@^w??A?N~_?????@??????H??????W_?????[G?????N@?????BwC?????^_G????@~?G????B~?C????B~_@????@???G????G@??_????_K?@????@?w?@????@@w??_????`{??G????G~??@????@Nw??C????D~_??G????N~???G??A??????C??@???_??@???O??W???G??A??F????_??G??{???@???O?Bw???@???O?Fw????_??G?F{????G??A?B~????@???O?~w????C??@?C??????G??A?G@?????G??A?GB?????C??@?CB_????@???O@@w?????G??A?G^??????_??G?b{?????@???O@Nw?????@???O@^w??????_??G?~{',
8937
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8938
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8939
'Q4': 'Or`HOm?OH@ABAG@C_POAJ',
8940
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8941
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8942
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8943
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8944
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8945
'Q5K03': '~?@_{S{ACbC_Q_f_?O?c?W_CA?SC?wCC?AAC?_b?CC_?OQ_?_f_???O???c???W_??CA???SC???wC??C?A??AC?_??b?C??C_?O??Q_?_??f??_?_???O?O?_?C?C?W??_?_C??A?A?S??C?C?w??C?CC???A?AAC???_?_b???C?CC_???O?OQ_???_?_f_???????O???????c???????W_??????CA???????SC???????wC??????C?A??????AC?_??????b?C??????C_?O??????Q_?_??????f??_?????_???O?????O?_?C?????C?W??_?????_C??A?????A?S??C?????C?w??C?????CC???A?????AAC???_?????_b???C?????CC_???O?????OQ_???_?????_f????_???_???????O???O???_???C???C???W????_???_??C????A???A???S????C???C???w????C???C??C?????A???A??AC?????_???_??b?????C???C??C_?????O???O??Q_?????_???_??f??????_???_?_???????O???O?O?_?????C???C?C?W??????_???_?_C??????A???A?A?S??????C???C?C?w??????C???C?CC???????A???A?AAC???????_???_?_b???????C???C?CC_???????O???O?OQ_???????_???_?_f',
8946
'Q6': '~?@?r`HOm?OH@ABAG@C_POAJ_?@??H??O_?KG?G@?@GC?D?G?J?GA??C@?_@?OO?GAB??_G_?@?PG?@?PO??_Gm?????O????H????@A????BA????G@????C_O???@OA????J?G???G??O???OG?O???OO?G???GK?A???AG??O???PG?@???@D??A???AJ??A??G????@??C??_??O?@??O??A??G?B???G??_?_???O?@?@G???O?@?@O???G??_?k???A??GA?????O?@?OG???@??C@@????A??GAB????A??GAG????@??C@C_????O?@?PO????A??GAJ',
8947
'Q7': '~?A?r`HOm?OH@ABAG@C_POAJ_?@??H??O_?KG?G@?@GC?D?G?J?GA??C@?_@?OO?GAB??_G_?@?PG?@?PO??_Gm?????O????H????@A????BA????G@????C_O???@OA????J?G???G??O???OG?O???OO?G???GK?A???AG??O???PG?@???@D??A???AJ??A??G????@??C??_??O?@??O??A??G?B???G??_?_???O?@?@G???O?@?@O???G??_?k???A??GA?????O?@?OG???@??C@@????A??GAB????A??GAG????@??C@C_????O?@?PO????A??GAJ_?????????@??????????H??????????O_?????????KG?????????G@?????????@GC?????????D?G?????????J?G????????A??C????????@?_@?????????OO?G????????AB??_????????G_?@?????????PG?@?????????PO??_????????Gk??G???????G????@???????@??G??C???????C?@???G???????G?B???G???????G?G???C???????C?C_??@???????@?@O???G???????G?J????_???????_G????@???????@?OG???@???????@?OO????_???????_GK????G???????GAG????@???????@?PG????C???????C@D?????G???????GAJ?????G????A??????????C????@?????_????@?????O????O?????G????A????B??????_????G????_?????@?????O???@G?????@?????O???@O??????_????G????k??????G????A???A???????@?????O???OG??????C????@???@@???????G????A???AB???????G????A???AG???????C????@???@C_??????@?????O???PO???????G????A???AJ????????_????G??_?????????@?????O?@??G???????@?????O?@??O????????_????G??_?K????????G????A??G?G????????@?????O?@?@G????????C????@??C?D?????????G????A??G?J?????????G????A??GA??????????C????@??C@?_????????@?????O?@?OO?????????G????A??GAB??????????_????G??_G_?????????@?????O?@?PG?????????@?????O?@?PO??????????_????G??_Gk',
8948
'W05': 'Dl{',
8949
'W06': 'Ehfw',
8950
'W07': 'FhENw',
8951
'W08': 'GhCKN{',
8952
'W09': 'HhCGKF~',
8953
'W10': 'IhCGGE@~w',
8954
'W11': 'JhCGGC@_N~_',
8955
'W12': 'KhCGGC@?K?~~',
8956
'W13': 'LhCGGC@?G?o@~~',
8957
'W14': 'MhCGGC@?G?_@_@~~_',
8958
'W15': 'NhCGGC@?G?_@?@_?~~w',
8959
'W16': 'OhCGGC@?G?_@?@??o?N~~',
8960
'W17': 'PhCGGC@?G?_@?@??_?K?@~~{',
8961
'W18': 'QhCGGC@?G?_@?@??_?G?@_?F~~w',
8962
'W19': 'RhCGGC@?G?_@?@??_?G?@??E??N~~w',
8963
'W20': 'ShCGGC@?G?_@?@??_?G?@??C??K??N~~{'}
8964
8965
for d in class0graphs_dict:
8966
g = Graph(class0graphs_dict[d])
8967
g.name(new = d)
8968
add_to_lists(g, class0graphs, all_graphs)
8969
8970
class0small = [g for g in class0graphs if g.order() < 30]
8971
for g in class0small:
8972
add_to_lists(g, problem_graphs)
8973
# Load the Sloane graphs
8974
def load_sloane_graphs():
8975
dc64_g6string ="~?@?JXxwm?OJ@wESEYMMbX{VDokGxAWvH[RkTAzA_Tv@w??wF]?oE\?OAHoC_@A@g?PGM?AKOQ??ZPQ?@rgt??{mIO?NSD_AD?mC\
8976
O?J?FG_FOOEw_FpGA[OAxa?VC?lWOAm_DM@?Mx?Y{A?XU?hwA?PM?PW@?G@sGBgl?Gi???C@_FP_O?OM?VMA_?OS?lSB??PS?`sU\
8977
??Gx?OyF_?AKOCN`w??PA?P[J??@C?@CU_??AS?AW^G??Ak?AwVZg|?Oy_@?????d??iDu???C_?D?j_???M??[Bl_???W??oEV?\
8978
???O??_CJNacABK?G?OAwP??b???GNPyGPCG@???"
8979
dc64 = Graph(dc64_g6string)
8980
dc64.name(new="dc64")
8981
add_to_lists(dc64, sloane_graphs, all_graphs)
8982
print "loaded graph dc64"
8983
8984
from cStringIO import StringIO
8985
from base64 import b64decode
8986
from gzip import GzipFile
8987
8988
dc1024_gzip = """
8989
H4sICD6oslkAA2RjMTAyNC50eHQArX2Lehs3zuijcSiONZZdZeT4oqRyS8nOWnETr+Lf2bXdbs+z
8990
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8991
/Ze+kn92kONC/ila8YcWz9NfxUR8Wgkl1p+EFKoxGJQS8D9IqTt42d6aP/tfBWR9gAIJ+l938L+j
8992
a/xrcYq5NGJobMbdqRZCTs4nmHe+hgxKvEw15vjPB0JmUS7mJ/h/1b/Siy/TH/h/+e3qlL5svy/p
8993
QX/FskWz+/wXJV62H+hBisXK09j5X0+3FK5QqV1NxGf/6TdHkxaEGSrUPtuUpmmWHvnl2j9OXjxW
8994
ceWQKdF5SnrR/senbtpAoZj8W4c/Hp/Dszrxj/KpZxm+6i58+VMHgprr7jSkWzzOXdm6aTYhj/l0
8995
/4H9tfnrmv21Xn1kf/U3W/aXWv6Po7lrf2V/Necfv7E/pf6yY39O9PUzT6xP/3IJzb+uZxWEgsQm
8996
ZL7ciOjrtRlf2C0WrtREi0kg7GTd9KYA4d7I+eLcjBREi7C4n/eCWkAjMerZjDSkh9DK9oe0PW1f
8997
iGvzT/kieslIoic93erpo6sRFvaHEg/UfHYcylUvXzTL1KsH3V4KlqafPOjpB9cK+L75spvQlPHU
8998
LD6L9RLKaVw2JA+etbKVEsvmaQtzWcrO5mt28nX32ROuJphDIlp15FMZIvslvny1dK7PxeZpRdXs
8999
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9000
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9001
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9002
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9003
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9004
Hqb2uSb2rwPH2LkJKqZn8HW+1pihCR/an/Dh5EZ/xTdbV7YwU0NC1Xpa7+5dIxk2+mR+m+bp0jaa
9005
dFlMxQ3GzizHiHyp/arU/42D7G+a3d061GS9ApQLN3E0+2K6w3xpibGorRsYZgbdnJvvsNBjZS5F
9006
IPr5TsEvW5rEws5t4BqdG4zNjGXaXin8pcEtexFaR8CygCIEtVv4NBU3t/D7p12WdgHhRCwuBckb
9007
vgFcrcxE0/Br1zzfV5KQ9n3gvX6J6wC7uG7E1k89v8T0ZsEx0oqUOz8Bt+GbqYv4tjEz0U7D5jzg
9008
bIGsa9OuU/fGjJ2bWzfHNAykUFu/CmjxGXvf8KPf7ItPoRLNmQBx6FE0T7Z6XnYwUxOa+WRt5qQb
9009
z+zjGj6K70pYochORUq42MDvh4347ZZobVae2/TqWcPsetZ2lW7EbBEoukC8V1JY9qrE1jdx18Cc
9010
7ZqrTl5ZOpptQDyB2dJPvveTf7uvPfsq8OvxbuJlDEPHjVJOasHFxi/OpraepSqc1woGv1v+O7bO
9011
06TXfsWAP4K8I2EwGHS7ya3lIOaPIHKAjACyqGkNJzE1YUiYNRI/y169eoFq7UeFkdhwmLSdnPYO
9012
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9013
TIpJX2eIdKHFRyeoAtcO31taxp9Fu3vwBcwCfeIL/l4I8bX9P/+SSYaS+KeZWfrCy6MNowBbH36P
9014
9IvvlIaVMHG/rfgReo0laNyvqfeDf9vxj0w2DeK+R6GjNqK08OIyfWlRhupp0z6KsLLpH/pH86bg
9015
kumR52+Tc8HBEzJ3a0EPjKyUZrL0f510PW8RX9ZVIKTbsLllAYgIknm3sDlxFXBppXgJjaN+uAdc
9016
G3w3/L/QaPLZExItr34LYBLdB54tVEjkZDcgoo3q4yjQzcXOv+jEf4XWaUmGstCzovkRV9khuvs9
9017
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9018
ic0txxCGPm24LSzu2YgVvk+bafdXeGmkpYRkSmb26uyFGcwyYeY9Su28ANWlqKhb3DafwIzotCWx
9019
Bpspx9XtCqlMgU5DYEt8Ttub4Gpj91wE96IppVLbf7MmNWyoVKQQt+2c/7kSyYy0pHm9BMGuXOZE
9020
fOWbYkBVwiatpIvQu39JiY9dVISRJjQTmuwshjn/OcoNO58CcUdbWFpDOolVzZv34waEv1tWLPDC
9021
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9022
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9023
duSWzbNNLiLQprq5pqTUyKrpulL76NZs36zI4DZou9JAFPqXazH/vzilLqecXYnTO1ddSl/gwFAz
9024
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9025
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9026
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9027
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9028
wiEfCCiMnmRdOEFFFOc3BC8wYLPsr7BaTFjxltwtzBmSVdn25RnerpK3wPxhL419xarU7mCAbjNC
9029
roAQYldskDPtV5R6AZOH9i1swZW/9NLuoSNKuheTmbptGigRm5XfhvNaypnZITa4aDWRwt3rxaIV
9030
ZzE3yGmoc9Gln2x7t4ljlOMC0aOObL4JBNK/bJ+icTzk/WOwLMTx2j4zGmE6/QsR/M2bC9aW1moG
9031
NKspri1YmSu+A1HNvGvOw+lIQLPVLSrs2D4P3pvN7szvwBg1Zl5tsWnvOR+T0H3PdkFh72FqTZB9
9032
Te4YlaDuf250Tg7MriNs3CN+YAHpn5zaL0oPywTpDHlPofJpqwv04OFCF/UU1dfMkPPGP3M8XzX+
9033
9hF38hrDpABDytUKfy8nfL+IMlefbEkV6raI/nREwL53tS31gDDLDfw+6l2aoS1nUCvIoC5vNM+A
9034
ukW7YiV1lqeASE4X36IMsFL/XugFk3nzG6L76yrKAENdPuc0SVSOwe/JehllgHfdS7Hj+gUcJvQ3
9035
t1EGrN+P+8LQMOTc3Jrfj0cvWY72x7yY42bzDfpO/Rnn8MpREfMVk7tdAKbZ+lqnNRdbde2fGaam
9036
h5MJqdmW1wrbMt3K05FCL21h2TJmED8rOkJJC9mBtJCcwQkaJbr7wz17UKitXcH69X/bOAvyq+6u
9037
lAVV8zA5l0kWEqRfZbLBoaOH5qnPV29AZ7DMdSoP9qT0lU1hUYOjHTNN0gVP4hIin82GO+hkwzcj
9038
P7S3vcwG2BZPEjt+kuhIgw1PN89JAy3h4vOspyVLR3lAA0R5YgYB52D96y/RgaSjzaqNZTrMJlDy
9039
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9040
sGHX00U6dmh78l8l/s4lGH2MmdofaaYLzHSnxMMyI6+XQFhvej3uCH2MZZ9sxOW3rCGU2OIOA5hK
9041
JF5IWBN7+aDd7j9qCTy+0upzPOzMEo4dd/0oftppF7XTEr8a4SIaEErsUFD4spR2iYzr7DXk8XDF
9042
1bAHIT49QRFuy4THFTrJJejoGE7MoMnxxIy25j3hCM3To0bY0SR5+ZH4s6SaJgoJUs+rxvS97t0r
9043
RsqUCBWzH3E30kxaiYdV65Lyz+e2+DsV64pxx9X08mfvlGl8RG22lsbJNqKxJTW9VnNvfsCz0Vpj
9044
yrmJstFWR857NffafU7kYmMRLVbRULyBRM2REYp2TlkU9RpuxHCl1tFgRKlwIf3pYFIciTLaTrNA
9045
JfaXhLn+3b9k5EhsSmnPgRkZ2BLHvT9YTMi0Zwfw74EPLtq9XGgjgf5JLAnkHSmtPkc3dpPuKjBl
9046
KoJM98Hp2Vo6k/JmmPyMGQeIeGB2tsZmqnxn+WhHonQ4zhTx0KRzQOC4zYSvhXSWaiYiWFE54B2x
9047
dCRou0m0hJB8p9lJqEgYnvt19gk2oyNNiXWkD0wyUmKcfpJlTIE3Hhcdfyq/TPONakjO0bFDJHH8
9048
IywgTM5gmz3WOlxvJR66xqFt2WsmRrDU9yyFDAtqw/Y31GUoouu1cHxZExYnt8EGMFQqHXLRGIj6
9049
HfealJIrqX8Lj6yBGi6ekp0MNgI7HBRzVjmGkJtENerZ9ybbqkjWQozcTcRZ0egGjy/ZkWKsXmcI
9050
44Wtt00XDp2FO2tO6W130SpwsrYknQQrOfEHFwxDSa2KswLbgvJmLOsdz8q6LWKNZA6E21c2Ys4i
9051
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9052
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9053
JpG8gnsxnHyXAf/lc6QyDRTpWSyXwM4A2UxYvNUrMyCIqDbcJKZazsFOQR6HvUp3najke5/5Kc1s
9054
N9Ahs7pu4q2YbwD9mJKtViiqBHFKftDR6VLoKRkfusEAQqm+ZyaUm8dIN8JabJEeuEvQZcGBSaC7
9055
v3YfsVwjhpg9orJHs5QEeI7q3N9dpudg4IlV8QkfLJoT3K2zPjm15pV57jbPLXHnznKv2p9x7tBo
9056
23QXD+oxNIDwTbONzxAFa6ltvqelHX2gbxarxzjp00xh1KE8uWLj6Xb6Ic4d6FomwjfwwCtcuMMg
9057
18t4tPA5ktMOwhHfcEqxLmznfd44u3DbeXo260eh3/0Ob1uSXl6VP5A0I2CSH/5546vLbMsucMu+
9058
c2j/UOaZ9G7uIGqGCzAKn34H05BRVTCcVG7QloatL35WKd6P/DtFFmYxWEpQ3k4GrbDaAyt8TbZm
9059
BK5q+UELrrL8qEmwLXmzNavhr0n2QP9jprwSVqvgaFkd9YJLrgTaFgfbf/bRnmK0O3/QqC4Xkb18
9060
VAGJlm2FCuC0tyPMLBXT/BTIUtqc9tngh0y//Ajs6cNa/Hqb5ncloQ1LVoMGjjV8KwUDuRKCbapc
9061
IAQnMYKrHIGrQt/FU8DWu70OVZh9FzNaKo5tqd+o3mS94xb8pjuJxIE+bNILmkJHK+kuYq0FFXv9
9062
GIbay5O3mGbgjq2WierMTbUvy9CP7W7CpdwopdNpBHAH8Mc7P9rgbOajyMBjEGUMsJV1GMRLyTrE
9063
dfpOxNPBHx3PnVVu14rP/yrZZ9gCVuHZ4kT8zU4+23Vfyp3i9zUSIqTsi9XoZuEKRD8TV+cZTwlJ
9064
8SA1Z6qbrVd1NGrbBCNGDnbW3yf1sKN62nn5BR7miwEUZ2UUR51s7SmzPDIorKFfR6NUnrv5jPs2
9065
q6lsQCPEkTmLI5uoBHZU4NlcpDkh6mRQxBgetFldigLY765EXjppUcAml4oJN5ZisDU9Dc+cvOMe
9066
tmbUYMdanN0Ei7q8Ks/hmVflQlsuJlDW+SM+tItx0BlrjuNMWy2pQAPPL+0f9bqQ6qJQF5RaCJ9Z
9067
Fs/kXQmHHR20ounw2o5LMCKnwQdH7BP9szTAbOETVjivzKljGwJFI38pLAF+2FNgvyAFXTnSmmRX
9068
lCTOWasOdZps+yZvry42sunDmN61MOJSJQTJ1pWxHmjIOTRSYg1rb4GSI3zOV1RHmcjFDUptdzp/
9069
wO5rgmjbLGEoMtLtIBDtZH2hnjWduohoL8yAGuw+e08aTGsr8CrIVj8+lE9TN4/Ze8xj9R+LJ1Oh
9070
X/C5PFIslk32HrEcOUEJlONDWKhd1tlrqpFda77pLlzLKQJ8RLO0GMgglbBLOMsm84PvaTqe/Cyv
9071
Eio/Tu2saJ46slqfvGQJWfLLfDCh+oX0RQ084OmNOLGTCN8f26MRn8ParZ02z53XilgAGUa5ROVd
9072
pCtVkUUQB6zr+cYSfEX3HoR4yCcLAbZwX57UMycbmS3SDCm5qKHBr6tc3gD0E4teozEEPE8zId0B
9073
dkDJCNHAuRMx+weUweLbo3n6LmcgSKAzW8Cjeax/bfjRvrBwWosnVn6/D2d/5vl7Sdzy9Eh3CyvF
9074
Mwnc2YhbBs/1aZ3zAat+LtZLOdWR+T3+L2A7eeDq7Kzgflbk5V5dBEvBNW5tX/sy27IVawsNBDqR
9075
i1CxDu9e+Qu6dv63OtrQWLFd6pnZUnY6Js6ZDHXDKwMpEJ+KIyiIc3R4Zej4fFMb0ITotozoiCOC
9076
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9077
S+5A5zlWZWM9r6PCQ064p/XRmTJWMMnbAr8jvZQfSWcbECEvUguVFJOuYGKi0BmaPXoryCJouBWT
9078
104yNRacMqMqfd78t147KPC8LDx51RXapsBVn2BAaRlncv/b70TBqk/KdER0IVF5J8xTdhWiBCNK
9079
dA+o+qxyAaK0VBSXBmkFbJ1d2hCq0qxsmAwmydAUbfbrgxO/FZcIHSSORpARDRy816ceFjmrFLEN
9080
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9081
Kn0Y6scmOrfjdEmGC7XmD3ARoS7oCJgAi9JrUqk5vBtFZ39Q784f8zT23IZDOAUieaDxuw6Rnx0P
9082
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9230
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9231
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9269
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9270
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9271
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9298
b2cbwSMUS40M8OH/Awl0aLwEVQEA
9299
"""
9300
f = GzipFile(mode='r', fileobj=StringIO(b64decode(dc1024_gzip)))
9301
dc1024_g6string = f.read()
9302
f.close()
9303
dc1024 = Graph(dc1024_g6string)
9304
dc1024.name(new="dc1024")
9305
add_to_lists(dc1024, sloane_graphs, all_graphs)
9306
print "loaded graph dc1024"
9307
9308
dc2048_gzip = """
9309
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9348
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10025
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10026
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10027
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10028
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10029
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10033
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10035
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10036
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10037
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10039
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10040
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10043
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10047
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10048
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10049
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10052
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10054
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10056
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10057
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10058
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10060
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10062
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10164
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10165
"""
10166
f = GzipFile(mode='r', fileobj=StringIO(b64decode(dc2048_gzip)))
10167
dc2048_g6string = f.read()
10168
f.close()
10169
dc2048 = Graph(dc2048_g6string)
10170
dc2048.name(new="dc2048")
10171
add_to_lists(dc2048, sloane_graphs, all_graphs)
10172
print "loaded graph dc2048"
10173
# Add all DIMACS graphs from the DIMACS subdirectory
10174
def load_dimacs_graphs():
10175
dimacs_g6_base64_dict = {'MANN_a81' : '''
10176
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10189
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10190
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10197
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10235
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10236
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10238
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10239
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10259
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10280
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10285
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10287
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10288
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10289
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10290
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10291
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10292
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10293
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10294
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10295
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10296
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10297
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10298
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10299
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10300
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10301
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10302
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10303
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10304
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10305
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10306
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10307
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10308
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10309
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10310
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10311
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10312
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10313
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10314
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10315
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10316
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10317
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10318
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10319
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10320
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10321
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10322
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10323
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10324
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10325
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10326
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10327
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10328
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10329
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10330
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10331
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10332
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10333
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10334
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10335
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10337
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10338
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10339
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10340
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10341
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10342
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10354
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10357
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10358
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10359
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10360
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10361
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10362
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10363
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10364
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10365
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10366
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10367
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10368
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10369
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10370
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10371
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10372
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10373
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10374
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10375
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10376
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10377
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10378
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10379
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10380
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10381
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10382
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10383
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10384
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10385
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10386
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10387
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10388
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10425
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10426
''',
10427
'gen400_p0.9_75' : '''
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10533
''',
10534
'brock200_4' : '''
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''',
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'p_hat300-2' : '''
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''',
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10801
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10802
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10803
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10807
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10808
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10809
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10810
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10811
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10812
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10813
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10814
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10815
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10816
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10817
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10818
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10819
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10820
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10821
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10822
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10823
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10824
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10825
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10826
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10827
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10828
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10829
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10830
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10831
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10832
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10833
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10834
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10835
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10836
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10837
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10838
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10839
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10840
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10841
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10842
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10843
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10844
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10845
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10846
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10847
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10848
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10849
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10850
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10851
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10852
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10853
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10854
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10855
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10856
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10857
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10858
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10859
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10860
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10861
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10862
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10863
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10865
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10866
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10867
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10868
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10869
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10870
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10873
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10874
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10877
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10878
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10879
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10880
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10881
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10882
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10883
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10886
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10887
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10888
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10889
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10890
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10891
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10892
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10893
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10894
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10895
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10896
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10897
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10898
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10899
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10900
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10901
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10902
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10903
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10904
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10905
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10906
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10907
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10908
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10909
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10910
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10911
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10912
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10913
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11044
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11049
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11060
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11061
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11062
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11063
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11064
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11065
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11066
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11067
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11068
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11069
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11070
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11071
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11072
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11073
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11074
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11075
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11076
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11077
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11078
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11079
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11080
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11081
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11082
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11083
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11084
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11085
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11086
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11087
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11088
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11089
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11090
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11091
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11092
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11093
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11094
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11095
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11096
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11097
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11098
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11099
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11100
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11101
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11102
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11103
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11104
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11105
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11106
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11107
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11108
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11109
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11110
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11111
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11112
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11113
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11114
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11115
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11116
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11117
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11118
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11119
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11120
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11121
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11122
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11123
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11124
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11125
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11126
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11127
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11128
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11129
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11130
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11131
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11132
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11133
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11134
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11135
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11136
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11137
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11138
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11139
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11140
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11141
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11142
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11143
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11144
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11145
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11146
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11147
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11148
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11149
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11150
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11151
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11152
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11153
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11154
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11155
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11157
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11158
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11159
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11160
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11161
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11162
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11163
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11164
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11165
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11166
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11167
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11168
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11169
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11170
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11171
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11329
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11330
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11332
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11333
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11334
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11336
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11337
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11338
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11339
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11340
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11341
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11342
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11343
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11344
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11345
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11346
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11347
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11348
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11350
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11351
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11352
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11353
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11354
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11356
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11357
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11358
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11359
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11360
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11361
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11362
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11363
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11364
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11365
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11366
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11367
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11368
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11369
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11370
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11412
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11413
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11422
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11423
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11425
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11426
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11749
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13301
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13801
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13809
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13812
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13813
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13814
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13815
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13816
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13817
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13819
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13820
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13823
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13886
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13891
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13894
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13899
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13900
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13901
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13902
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13903
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13904
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13905
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13909
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13910
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13911
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13912
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13913
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13914
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13915
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13916
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13917
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13918
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13919
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13920
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13921
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13922
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13923
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13924
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13925
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13926
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13927
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13928
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13929
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13930
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13931
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13932
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13933
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13934
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13935
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13936
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13937
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13938
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13939
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13940
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13941
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13942
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13943
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13944
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13945
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13946
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13947
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13948
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13949
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13950
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13951
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13952
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13953
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13954
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13955
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13956
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13957
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13958
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13959
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13960
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13961
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13962
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13963
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13964
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13965
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13966
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13967
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13968
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13969
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13970
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13971
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13972
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13973
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13974
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13975
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13976
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13977
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13978
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13979
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13981
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13982
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13983
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13984
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13985
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13986
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13987
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13988
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13989
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13990
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13991
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13992
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13993
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13994
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13995
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13996
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13997
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13998
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13999
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14000
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14001
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14002
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14003
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14004
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14005
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14006
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14007
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14008
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14009
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14010
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14011
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14012
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14013
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14014
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14170
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14178
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14179
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14180
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14181
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14182
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14183
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14184
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14185
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14186
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14187
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14188
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14189
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14190
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14191
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14192
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14193
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14194
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14195
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14196
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14197
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14198
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14199
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14200
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14201
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14202
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14203
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14204
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14205
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14206
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14207
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14208
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14209
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14210
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14211
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14212
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14213
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14214
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14215
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14216
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14217
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14218
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14219
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14221
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14224
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14225
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14230
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14260
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14262
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14264
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14267
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14268
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14270
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14271
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14272
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14280
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14296
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14297
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14298
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14299
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14300
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14301
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14302
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14303
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14304
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14305
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14306
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14307
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14308
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14309
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14310
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14311
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14312
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14313
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14314
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14315
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14316
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14317
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14318
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14319
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14320
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14321
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14322
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14323
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14324
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14325
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14326
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14327
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14328
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14329
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14330
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14331
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14332
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14333
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14334
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14335
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14336
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14337
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14338
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14339
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14340
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14341
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14342
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14343
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14344
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14347
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14348
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14350
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14351
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14352
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14353
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14354
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14355
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14356
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14358
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14359
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14360
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14362
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14363
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14364
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14367
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14368
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14369
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14371
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14372
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14373
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14374
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14377
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14378
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14379
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14380
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14381
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14382
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14383
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14384
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14385
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14386
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14387
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14388
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14389
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14390
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14391
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14392
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14393
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14394
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14395
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14396
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14397
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14398
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14399
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14400
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14965
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14969
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14983
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''',
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''',
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16266
''',
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'gen200_p0.9_44' : '''
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16296
''',
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'gen400_p0.9_55' : '''
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''',
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16893
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16895
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16896
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16897
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16898
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16899
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16900
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16901
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16902
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16903
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16904
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16905
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16906
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16907
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16908
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16909
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16910
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16911
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16912
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16913
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16914
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16915
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16916
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16917
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16918
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16919
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16920
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16921
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16922
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16923
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16924
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16925
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16926
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16927
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16928
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16929
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16930
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16931
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16932
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16933
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16934
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16935
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16936
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16937
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16938
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16985
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16987
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16989
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16991
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17028
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17030
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17031
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17032
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17033
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17034
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17035
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17036
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17037
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17038
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17039
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17040
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17041
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17042
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17043
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17044
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17045
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17046
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17050
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17051
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17053
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17054
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17055
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17057
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17058
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17060
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''',
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'keller5' : '''
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''',
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17939
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17946
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17947
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17948
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17950
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17951
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17960
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17962
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17963
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17964
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17965
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17966
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17967
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17968
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17969
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17970
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17971
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17972
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17973
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17974
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17975
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17976
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17986
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18002
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18019
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18020
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18022
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18026
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18028
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18034
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18460
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18464
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18465
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18469
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18476
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18485
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18486
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19587
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19588
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19589
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19590
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19591
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19592
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19593
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19594
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19595
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19596
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19597
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19598
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19599
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19600
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19601
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19602
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19603
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19604
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19605
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19606
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19607
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19608
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19609
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19610
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19611
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19612
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19613
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19614
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19615
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19616
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19617
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19618
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19619
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19620
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19621
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19622
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19623
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19624
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19625
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19626
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19627
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19628
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19629
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19630
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19631
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19632
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19633
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19634
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19635
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19636
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19637
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19638
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19639
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19640
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19641
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19642
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19643
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19644
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19645
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19646
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19647
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19648
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19649
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19650
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19651
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19652
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19653
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19654
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19655
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19656
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19657
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19658
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19659
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19660
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19661
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19662
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19663
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19664
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19665
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19666
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19667
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19668
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19669
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19670
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19671
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19672
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19673
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19674
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19675
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19676
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19677
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19678
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19679
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19680
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19681
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19682
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19683
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19684
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19685
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19686
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19687
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19688
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19689
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19690
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19691
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19692
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19693
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19694
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19695
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19696
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19697
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19698
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19699
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19700
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19701
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19702
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19703
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19704
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19705
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19706
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19707
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19708
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19709
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19710
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19711
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19712
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19713
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19714
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19715
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19716
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19717
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19718
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19719
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19720
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19721
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19722
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19723
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19724
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19725
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19726
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19727
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19728
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19729
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19730
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19731
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19732
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19733
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19734
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19735
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19736
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19737
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19738
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19739
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19740
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19741
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19742
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19743
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19744
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19745
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19746
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19747
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19748
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19749
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19750
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19751
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19752
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19753
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19754
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19755
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19756
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19757
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19758
VozIm+vRr4YDm7/L1EeoaadllgTOrbNrGKXBF3OENP+YfCs1t2OtQw0OQpeH89pwZdUNmntVvbK7
19759
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19760
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19761
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19762
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19763
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19764
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19765
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19766
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19767
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19768
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19769
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19770
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19771
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19772
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19773
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19774
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19775
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19776
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19777
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19778
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19779
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19780
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19781
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19782
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19783
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19784
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19785
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19786
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19787
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19788
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19789
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19790
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19791
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19792
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19793
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19794
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19795
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19796
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19797
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19798
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19799
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19800
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19801
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19802
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19803
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19804
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19805
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19806
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19807
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19808
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19809
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19810
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19811
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19812
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19813
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19814
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19815
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19816
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19817
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19818
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19819
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19820
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19821
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19822
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19823
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19824
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19825
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19826
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19827
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19828
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19829
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19830
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19831
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19832
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19833
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19834
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19835
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19836
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19837
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19838
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19839
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19840
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19841
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19842
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19843
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19844
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19845
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19846
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19847
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19848
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19849
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19850
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19851
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19852
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19853
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19854
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19855
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19856
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19857
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19858
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19859
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19860
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19861
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19862
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19863
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19864
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19865
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19866
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19867
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19868
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19869
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19870
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19871
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19872
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19873
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19874
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19875
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19876
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19877
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19878
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19879
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19880
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19881
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19882
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19883
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19884
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19885
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19886
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19887
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19888
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19889
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19890
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19891
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19892
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19893
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19894
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19895
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19896
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19897
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19898
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19899
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19900
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19901
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19902
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19903
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19904
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19905
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19906
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19907
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19908
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19909
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19910
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19911
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19912
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19913
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19914
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19915
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19916
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19917
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19918
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19919
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19920
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19921
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19922
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19923
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19924
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19925
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19926
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19927
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19928
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19929
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19930
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19931
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19932
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19933
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19934
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19935
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19936
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19937
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19938
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19939
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19940
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19941
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19942
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19943
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19944
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19945
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19946
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19947
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19948
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19949
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19950
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19951
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19952
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19953
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19954
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19955
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19956
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19957
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19958
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19959
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19960
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19961
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19962
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19963
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19964
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19965
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19966
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19967
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19968
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19969
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19970
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19971
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19972
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19973
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20790
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20795
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20801
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20804
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20805
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20806
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20807
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20808
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20809
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20810
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20811
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20812
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20817
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''',
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26743
''',
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'hamming10-4' : '''
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''',
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''',
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'p_hat1500-3' : '''
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31976
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31977
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31978
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31979
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31980
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31981
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31982
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31983
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31984
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31985
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31986
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31987
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31988
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31989
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31990
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31991
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31992
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31993
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31994
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31995
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31996
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31997
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'DSJC500_5' : '''
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35904
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35905
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35909
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35910
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35911
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35912
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35913
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35914
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35915
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35916
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35917
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35918
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35919
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35920
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35921
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35922
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35923
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35924
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35925
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35926
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35927
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35928
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35929
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35930
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35931
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35932
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35935
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35937
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35938
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35939
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35940
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35943
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38738
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38739
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38741
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38742
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38743
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38744
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38745
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38747
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38748
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38749
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38750
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38751
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38752
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38754
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38755
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38758
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38760
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38762
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38764
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38765
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38766
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38767
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38769
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38770
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38771
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38772
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38773
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38774
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38775
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38776
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38780
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38781
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38806
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38810
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38816
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38817
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38818
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38819
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38821
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38823
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38824
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38825
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38827
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38828
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38829
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38831
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38832
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38833
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38834
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38835
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42000
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42001
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42009
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42011
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42020
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42021
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42022
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42023
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42024
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42025
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42026
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42027
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42028
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42029
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42030
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42031
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42032
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42040
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42041
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42042
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42044
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42045
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42048
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42049
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42056
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42057
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42059
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42060
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42061
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42062
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42063
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42064
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49912
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51175
''',
51176
'gen200_p0.9_55' : '''
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DAAA
51205
''',
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'p_hat1500-2' : '''
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51901
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51903
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51907
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51908
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51910
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51911
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51912
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51915
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51916
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51917
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51918
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51919
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51920
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51921
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51922
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51923
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51924
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51925
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51926
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51928
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51929
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51930
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51931
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51933
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51934
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51937
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51967
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51969
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51971
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51977
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51978
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51979
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51980
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51981
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51982
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51983
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51984
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51985
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51986
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51987
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51988
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51989
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51990
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51991
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51992
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51993
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53667
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53668
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53669
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53670
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53671
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53675
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53676
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53677
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53679
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53688
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53689
D12zT13z2wIA
53690
'''}
53691
from cStringIO import StringIO
53692
from base64 import b64decode
53693
from gzip import GzipFile
53694
53695
for name in dimacs_g6_base64_dict:
53696
f = GzipFile(mode='r', fileobj=StringIO(b64decode(dimacs_g6_base64_dict[name])))
53697
g6string = f.read()
53698
f.close()
53699
g = Graph(g6string)
53700
g.name(new = name)
53701
add_to_lists(g, dimacs_graphs, all_graphs)
53702
print "loaded graph - ", g
53703
53704
print("loaded graphs")
53705
53706
#############################################################################
53707
# End of graphs section #
53708
#############################################################################
53709
53710
print("\nRemember to load DIMACS and Sloane graphs if you want them")
53711
53712
#############################################################################
53713
# End of gt.sage file #
53714
#############################################################################
53715
53716