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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):
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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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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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count = 0
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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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count += 1
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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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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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n = g.order()
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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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V = g.vertices()
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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 i in [1..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([len_v..len_v+3])
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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 A[u][v] == 1 and A[u][w] == 1 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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#TESTING
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#check for invariant relation that separtates G from class defined by property
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def find_separating_invariant_relation(g, objects, property, invariants):
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L = [x for x in objects if (property)(x)]
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for inv1 in invariants:
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for inv2 in invariants:
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if inv1(g) > inv2(g) and all(inv1(x) <= inv2(x) for x in L):
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return inv1.__name__, inv2.__name__
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print "no separating invariants"
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#finds "difficult" graphs for necessary conditions, finds graphs which don't have property but which have all necessary conditions
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def test_properties_upper_bound_theory(objects, property, theory):
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for g in objects:
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if not property(g) and all(f(g) for f in theory):
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print g.name()
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#finds "difficult" graphs for sufficient conditions, finds graphs which dont have any sufficient but do have property
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def test_properties_lower_bound_theory(objects, property, theory):
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for g in objects:
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if property(g) and not any(f(g) for f in theory):
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print g.name()
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def find_coextensive_properties(objects, properties):
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for p1 in properties:
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for p2 in properties:
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if p1 != p2 and all(p1(g) == p2(g) for g in objects):
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print p1.__name__, p2.__name__
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print "DONE!"
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print("loaded utilities")
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#############################################################################
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# End of utilities section #
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#############################################################################
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#GRAPH INVARIANTS
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all_invariants = []
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efficient_invariants = []
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intractable_invariants = []
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theorem_invariants = []
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broken_invariants = []
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def distinct_degrees(g):
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"""
419
returns the number of distinct degrees of a graph
420
sage: distinct_degrees(p4)
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2
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sage: distinct_degrees(k4)
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1
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"""
425
return len(set(g.degree()))
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add_to_lists(distinct_degrees, efficient_invariants, all_invariants)
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#inspired by the Friendship Theorem
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def common_neighbors(g,v,w):
430
"""
431
returns the Set of common neighbors of v and w in graph g
432
sage: common_neighbors(p4,0,3)
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{}
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sage: common_neighbors(p4,0,2)
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{1}
436
"""
437
Nv = Set(g.neighbors(v))
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Nw = Set(g.neighbors(w))
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return Nv.intersection(Nw)
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441
def max_common_neighbors(g):
442
"""
443
returns the maximum number of common neighbors of any pair of distinct vertices in g
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sage: max_common_neighbors(p4)
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1
446
sage: max_common_neighbors(k4)
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2
448
"""
449
max = 0
450
V = g.vertices()
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n = g.order()
452
for i in range(n):
453
for j in range(n):
454
if i < j:
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temp = len(common_neighbors(g, V[i], V[j]))
456
if temp > max:
457
max = temp
458
return max
459
add_to_lists(max_common_neighbors, efficient_invariants, all_invariants)
460
461
def min_common_neighbors(g):
462
"""
463
returns the minimum number of common neighbors of any pair of distinct vertices in g,
464
which is necessarily 0 for disconnected graphs
465
sage: min_common_neighbors(p4)
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0
467
sage: min_common_neighbors(k4)
468
2
469
"""
470
n = g.order()
471
min = n
472
V = g.vertices()
473
for i in range(n):
474
for j in range(n):
475
if i < j:
476
temp = len(common_neighbors(g, V[i], V[j]))
477
#if temp == 0:
478
#print "i={}, j={}".format(i,j)
479
if temp < min:
480
min = temp
481
return min
482
add_to_lists(min_common_neighbors, efficient_invariants, all_invariants)
483
484
def mean_common_neighbors(g):
485
"""
486
returns the average number of common neighbors of any pair of distinct vertices in g
487
sage: mean_common_neighbors(p4)
488
1/3
489
sage: mean_common_neighbors(k4)
490
2
491
"""
492
V = g.vertices()
493
n = g.order()
494
sum = 0
495
for i in range(n):
496
for j in range(n):
497
if i < j:
498
sum += len(common_neighbors(g, V[i], V[j]))
499
return 2*sum/(n*(n-1))
500
add_to_lists(mean_common_neighbors, efficient_invariants, all_invariants)
501
502
def min_degree(g):
503
"""
504
Returns the minimum of all degrees of the graph g.
505
506
sage: min_degree(graphs.CompleteGraph(5))
507
4
508
sage: min_degree(graphs.CycleGraph(5))
509
2
510
sage: min_degree(graphs.StarGraph(5))
511
1
512
sage: min_degree(graphs.CompleteBipartiteGraph(3,5))
513
3
514
"""
515
return min(g.degree())
516
add_to_lists(min_degree, efficient_invariants, all_invariants)
517
518
def max_degree(g):
519
"""
520
Returns the maximum of all degrees of the graph g.
521
522
sage: max_degree(graphs.CompleteGraph(5))
523
4
524
sage: max_degree(graphs.CycleGraph(5))
525
2
526
sage: max_degree(graphs.StarGraph(5))
527
5
528
sage: max_degree(graphs.CompleteBipartiteGraph(3,5))
529
5
530
"""
531
return max(g.degree())
532
add_to_lists(max_degree, efficient_invariants, all_invariants)
533
534
def median_degree(g):
535
return median(g.degree())
536
add_to_lists(median_degree, efficient_invariants, all_invariants)
537
538
def inverse_degree(g):
539
return sum([(1.0/d) for d in g.degree() if d!= 0])
540
add_to_lists(inverse_degree, efficient_invariants, all_invariants)
541
542
def eulerian_faces(g):
543
"""
544
Returns 2 - order + size, which is the number of faces if the graph is planar,
545
a consequence of Euler's Formula
546
547
sage: eulerian_faces(graphs.CycleGraph(5))
548
2
549
sage: eulerian_faces(graphs.DodecahedralGraph())
550
12
551
"""
552
n = g.order()
553
m = g.size()
554
return 2 - n + m
555
add_to_lists(eulerian_faces, efficient_invariants, all_invariants)
556
557
def barrus_q(g):
558
"""
559
If the degrees sequence is in non-increasing order, with index starting at 1,
560
barrus_q = max(k:d_k >= k)
561
562
Defined by M. Barrus in "Havel-Hakimi Residues of Unigraphs", 2012
563
564
sage: barrus_q(graphs.CompleteGraph(5))
565
4
566
sage: barrus_q(graphs.StarGraph(3))
567
1
568
569
"""
570
Degrees = g.degree()
571
Degrees.sort()
572
Degrees.reverse()
573
return max(k for k in range(g.order()) if Degrees[k] >= (k+1)) + 1
574
add_to_lists(barrus_q, efficient_invariants, all_invariants)
575
576
def barrus_bound(g):
577
"""
578
returns n - barrus q
579
defined in: Barrus, Michael D. "Havel–Hakimi residues of unigraphs." Information Processing Letters 112.1 (2012): 44-48.
580
sage: barrus_bound(k4)
581
1
582
sage: barrus_bound(graphs.OctahedralGraph())
583
2
584
"""
585
return g.order() - barrus_q(g)
586
add_to_lists(barrus_bound, efficient_invariants, all_invariants)
587
588
def matching_number(g):
589
"""
590
Returns the matching number of the graph g, i.e., the size of a maximum
591
matching. A matching is a set of independent edges.
592
593
sage: matching_number(graphs.CompleteGraph(5))
594
2
595
sage: matching_number(graphs.CycleGraph(5))
596
2
597
sage: matching_number(graphs.StarGraph(5))
598
1
599
sage: matching_number(graphs.CompleteBipartiteGraph(3,5))
600
3
601
"""
602
return int(g.matching(value_only=True, use_edge_labels=False))
603
add_to_lists(matching_number, efficient_invariants, all_invariants)
604
605
def residue(g):
606
"""
607
If the Havel-Hakimi process is iterated until a sequence of 0s is returned,
608
residue is defined to be the number of zeros of this sequence.
609
610
sage: residue(k4)
611
1
612
sage: residue(p4)
613
2
614
615
"""
616
seq = g.degree_sequence()
617
618
while seq[0] > 0:
619
d = seq.pop(0)
620
seq[:d] = [k-1 for k in seq[:d]]
621
seq.sort(reverse=True)
622
623
return len(seq)
624
add_to_lists(residue, efficient_invariants, all_invariants)
625
626
def annihilation_number(g):
627
seq = sorted(g.degree())
628
629
a = 0
630
while sum(seq[:a+1]) <= sum(seq[a+1:]):
631
a += 1
632
633
return a
634
add_to_lists(annihilation_number, efficient_invariants, all_invariants)
635
636
def fractional_alpha(g):
637
if len(g.vertices()) == 0:
638
return 0
639
p = MixedIntegerLinearProgram(maximization=True)
640
x = p.new_variable(nonnegative=True)
641
p.set_objective(sum(x[v] for v in g.vertices()))
642
643
for v in g.vertices():
644
p.add_constraint(x[v], max=1)
645
646
for (u,v) in g.edge_iterator(labels=False):
647
p.add_constraint(x[u] + x[v], max=1)
648
649
return p.solve()
650
add_to_lists(fractional_alpha, efficient_invariants, all_invariants)
651
652
def fractional_covering(g):
653
if len(g.vertices()) == 0:
654
return 0
655
p = MixedIntegerLinearProgram(maximization=False)
656
x = p.new_variable(nonnegative=True)
657
p.set_objective(sum(x[v] for v in g.vertices()))
658
659
for v in g.vertices():
660
p.add_constraint(x[v], min=1)
661
662
for (u,v) in g.edge_iterator(labels=False):
663
p.add_constraint(x[u] + x[v], min=1)
664
665
return p.solve()
666
add_to_lists(fractional_covering, efficient_invariants, all_invariants)
667
668
def cvetkovic(g):
669
eigenvalues = g.spectrum()
670
positive = 0
671
negative = 0
672
zero = 0
673
for e in eigenvalues:
674
if e > 0:
675
positive += 1
676
elif e < 0:
677
negative += 1
678
else:
679
zero += 1
680
681
return zero + min([positive, negative])
682
add_to_lists(cvetkovic, efficient_invariants, all_invariants)
683
684
def cycle_space_dimension(g):
685
return g.size()-g.order()+g.connected_components_number()
686
add_to_lists(cycle_space_dimension, efficient_invariants, all_invariants)
687
688
def card_center(g):
689
return len(g.center())
690
add_to_lists(card_center, efficient_invariants, all_invariants)
691
692
def card_periphery(g):
693
return len(g.periphery())
694
add_to_lists(card_periphery, efficient_invariants, all_invariants)
695
696
def max_eigenvalue(g):
697
return max(g.adjacency_matrix().change_ring(RDF).eigenvalues())
698
add_to_lists(max_eigenvalue, efficient_invariants, all_invariants)
699
700
def min_eigenvalue(g):
701
return min(g.adjacency_matrix().change_ring(RDF).eigenvalues())
702
add_to_lists(min_eigenvalue, efficient_invariants, all_invariants)
703
704
def resistance_distance_matrix(g):
705
L = g.laplacian_matrix()
706
n = g.order()
707
J = ones_matrix(n,n)
708
temp = L+(1.0/n)*J
709
X = temp.inverse()
710
R = (1.0)*ones_matrix(n,n)
711
for i in range(n):
712
for j in range(n):
713
R[i,j] = X[i,i] + X[j,j] - 2*X[i,j]
714
return R
715
716
def kirchhoff_index(g):
717
R = resistance_distance_matrix(g)
718
return .5*sum(sum(R))
719
add_to_lists(kirchhoff_index, efficient_invariants, all_invariants)
720
721
def largest_singular_value(g):
722
A = matrix(RDF,g.adjacency_matrix())
723
svd = A.SVD()
724
sigma = svd[1]
725
return sigma[0,0]
726
add_to_lists(largest_singular_value, efficient_invariants, all_invariants)
727
728
def card_max_cut(g):
729
return g.max_cut(value_only=True)
730
add_to_lists(card_max_cut, intractable_invariants, all_invariants)
731
732
def welsh_powell(g):
733
"""
734
for degrees d_1 >= ... >= d_n
735
returns the maximum over all indices i of of the min(i,d_i + 1)
736
737
sage: welsh_powell(k5) = 4
738
"""
739
n= g.order()
740
D = g.degree()
741
D.sort(reverse=True)
742
mx = 0
743
for i in range(n):
744
temp = min({i,D[i]})
745
if temp > mx:
746
mx = temp
747
return mx + 1
748
add_to_lists(welsh_powell, efficient_invariants, all_invariants)
749
750
#outputs upper bound from Brooks Theorem: returns Delta + 1 for complete and odd cycles
751
def brooks(g):
752
Delta = max(g.degree())
753
delta = min(g.degree())
754
n = g.order()
755
if is_complete(g):
756
return Delta + 1
757
elif n%2 == 1 and g.is_connected() and Delta == 2 and delta == 2: #same as if all degrees are 2
758
return Delta + 1
759
else:
760
return Delta
761
add_to_lists(brooks, efficient_invariants, all_invariants)
762
763
#wilf's upper bound for chromatic number
764
def wilf(g):
765
return max_eigenvalue(g) + 1
766
add_to_lists(wilf, efficient_invariants, all_invariants)
767
768
#a measure of irregularity
769
def different_degrees(g):
770
return len(set(g.degree()))
771
add_to_lists(different_degrees, efficient_invariants, all_invariants)
772
773
def szekeres_wilf(g):
774
"""
775
Returns 1+ max of the minimum degrees for all subgraphs
776
Its an upper bound for chromatic number
777
778
sage: szekeres_wilf(k5)
779
5
780
"""
781
#removes a vertex, if possible, of degree <= i
782
def remove_vertex_of_degree(gc,i):
783
Dc = gc.degree()
784
V = gc.vertices()
785
#print "Dc is %s, V is %s" %(Dc,V)
786
mind = min(Dc)
787
#print "mind is %s" %mind
788
if mind <= i:
789
790
ind = Dc.index(mind)
791
#print "ind is %s, vertex is %s" %(ind,V[ind])
792
return gc.delete_vertex(V[ind])
793
else:
794
return gc
795
D = g.degree()
796
delta = min(D)
797
Delta = max(D)
798
for i in range(delta,Delta+1):
799
gc = copy(g)
800
value = g.order() + 1
801
while gc.size() > 0 and gc.order() < value:
802
#gc.show()
803
value = gc.order()
804
remove_vertex_of_degree(gc,i)
805
if gc.size() == 0:
806
return i + 1
807
add_to_lists(szekeres_wilf, efficient_invariants, all_invariants)
808
809
def average_vertex_temperature(g):
810
D = g.degree()
811
n = g.order()
812
return sum([D[i]/(n-D[i]+0.0) for i in range(n)])/n
813
add_to_lists(average_vertex_temperature, efficient_invariants, all_invariants)
814
815
def sum_temperatures(g):
816
D = g.degree()
817
n = g.order()
818
return sum([D[i]/(n-D[i]+0.0) for i in range(n)])
819
add_to_lists(sum_temperatures, efficient_invariants, all_invariants)
820
821
def randic(g):
822
D = g.degree()
823
V = g.vertices()
824
if min(D) == 0:
825
return oo
826
sum = 0
827
for e in g.edges():
828
v = e[0]
829
i = V.index(v)
830
w = e[1]
831
j = V.index(w)
832
sum += 1.0/(D[i]*D[j])**0.5
833
return sum
834
add_to_lists(randic, efficient_invariants, all_invariants)
835
836
#a very good lower bound for alpha
837
def max_even_minus_even_horizontal(g):
838
"""
839
finds def max_even_minus_even_horizontal for each component and adds them up.
840
"""
841
mx_even=0
842
Gcomps=g.connected_components_subgraphs()
843
844
while Gcomps != []:
845
H=Gcomps.pop()
846
temp=max_even_minus_even_horizontal_component(H)
847
mx_even+=temp
848
#print "temp = {}, mx_even = {}".format(temp,mx_even)
849
850
return mx_even
851
add_to_lists(max_even_minus_even_horizontal, efficient_invariants, theorem_invariants, all_invariants)
852
853
def max_even_minus_even_horizontal_component(g):
854
"""
855
for each vertex v, find the number of vertices at even distance from v,
856
and substract the number of edges induced by these vertices.
857
this number is a lower bound for independence number.
858
take the max. returns 0 if graph is not connected
859
"""
860
if g.is_connected()==False:
861
return 0
862
863
distances = g.distance_all_pairs()
864
mx=0
865
n=g.order()
866
for v in g.vertices():
867
Even=[]
868
for w in g.vertices():
869
if distances[v][w]%2==0:
870
Even.append(w)
871
872
#print len(Even), len(g.subgraph(Even).edges())
873
l=len(Even)-len(g.subgraph(Even).edges())
874
if l>mx:
875
mx=l
876
return mx
877
878
def laplacian_energy(g):
879
L = g.laplacian_matrix().change_ring(RDF).eigenvalues()
880
Ls = [1/lam**2 for lam in L if lam > 0]
881
return 1 + sum(Ls)
882
add_to_lists(laplacian_energy, efficient_invariants, all_invariants)
883
884
def laplacian_energy_like(g):
885
"""
886
Returns the sum of the square roots of the laplacian eigenvalues
887
888
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.
889
"""
890
return sum([sqrt(x) for x in g.spectrum(laplacian = True)])
891
add_to_lists(laplacian_energy_like, efficient_invariants, all_invariants)
892
893
#sum of the positive eigenvalues of a graph
894
def gutman_energy(g):
895
L = g.adjacency_matrix().change_ring(RDF).eigenvalues()
896
Ls = [lam for lam in L if lam > 0]
897
return sum(Ls)
898
add_to_lists(gutman_energy, efficient_invariants, all_invariants)
899
900
#the second smallest eigenvalue of the Laplacian matrix of a graph, also called the "algebraic connectivity" - the smallest should be 0
901
def fiedler(g):
902
L = g.laplacian_matrix().change_ring(RDF).eigenvalues()
903
L.sort()
904
return L[1]
905
add_to_lists(fiedler, broken_invariants, all_invariants)
906
907
def degree_variance(g):
908
mu = mean(g.degree())
909
s = sum((x-mu)**2 for x in g.degree())
910
return s/g.order()
911
add_to_lists(degree_variance, efficient_invariants, all_invariants)
912
913
def graph_rank(g):
914
return g.adjacency_matrix().rank()
915
add_to_lists(graph_rank, efficient_invariants, all_invariants)
916
917
def card_positive_eigenvalues(g):
918
return len([lam for lam in g.adjacency_matrix().eigenvalues() if lam > 0])
919
add_to_lists(card_positive_eigenvalues, efficient_invariants, all_invariants)
920
921
def card_zero_eigenvalues(g):
922
return len([lam for lam in g.adjacency_matrix().eigenvalues() if lam == 0])
923
add_to_lists(card_zero_eigenvalues, efficient_invariants, all_invariants)
924
925
def card_negative_eigenvalues(g):
926
return len([lam for lam in g.adjacency_matrix().eigenvalues() if lam < 0])
927
add_to_lists(card_negative_eigenvalues, efficient_invariants, all_invariants)
928
929
def card_cut_vertices(g):
930
return len((g.blocks_and_cut_vertices())[1])
931
add_to_lists(card_cut_vertices, efficient_invariants, all_invariants)
932
933
def card_connectors(g):
934
return g.order() - card_cut_vertices(g)
935
add_to_lists(card_connectors, efficient_invariants, all_invariants)
936
937
#return number of leafs or pendants
938
def card_pendants(g):
939
return sum([x for x in g.degree() if x == 1])
940
add_to_lists(card_pendants, efficient_invariants, all_invariants)
941
942
def vertex_con(g):
943
return g.vertex_connectivity()
944
add_to_lists(vertex_con, efficient_invariants, all_invariants)
945
946
def edge_con(g):
947
return g.edge_connectivity()
948
add_to_lists(edge_con, efficient_invariants, all_invariants)
949
950
#returns number of bridges in graph
951
def card_bridges(g):
952
gs = g.strong_orientation()
953
bridges = []
954
for scc in gs.strongly_connected_components():
955
bridges.extend(gs.edge_boundary(scc))
956
return len(bridges)
957
add_to_lists(card_bridges, efficient_invariants, all_invariants)
958
959
#upper bound for the domination number
960
def alon_spencer(g):
961
delta = min(g.degree())
962
n = g.order()
963
return n*((1+log(delta + 1.0)/(delta + 1)))
964
add_to_lists(alon_spencer, efficient_invariants, all_invariants)
965
966
#lower bound for residue and, hence, independence number
967
def caro_wei(g):
968
return sum([1.0/(d + 1) for d in g.degree()])
969
add_to_lists(caro_wei, efficient_invariants, all_invariants)
970
971
#equals 2*size, the 1st theorem of graph theory
972
def degree_sum(g):
973
return sum(g.degree())
974
add_to_lists(degree_sum, efficient_invariants, all_invariants)
975
976
#smallest sum of degrees of non-adjacent degrees, invariant in ore condition for hamiltonicity
977
#default for complete graph?
978
def sigma_2(g):
979
if g.size() == g.order()*(g.order()-1)/2:
980
return g.order()
981
Dist = g.distance_all_pairs()
982
return min(g.degree(v) + g.degree(w) for v in g for w in g if Dist[v][w] > 1)
983
add_to_lists(sigma_2, efficient_invariants, all_invariants)
984
985
#cardinality of the automorphism group of the graph
986
def order_automorphism_group(g):
987
return g.automorphism_group(return_group=False, order=True)
988
add_to_lists(order_automorphism_group, efficient_invariants, all_invariants)
989
990
#in sufficient condition for graphs where vizing's independence theorem holds
991
def brinkmann_steffen(g):
992
E = g.edges()
993
if len(E) == 0:
994
return 0
995
Dist = g.distance_all_pairs()
996
return min(g.degree(v) + g.degree(w) for v in g for w in g if Dist[v][w] == 1)
997
add_to_lists(brinkmann_steffen, efficient_invariants, all_invariants)
998
999
def alpha_critical_optimum(g, alpha_critical_names):
1000
1001
n = g.order()
1002
V = g.vertices()
1003
#g.show()
1004
1005
alpha_critical_graph_names = []
1006
1007
#get alpha_critical graphs with order <= n
1008
for name in alpha_critical_names:
1009
h = Graph(name)
1010
if h.order() <= n:
1011
alpha_critical_graph_names.append(h.graph6_string())
1012
1013
#print alpha_critical_graphs
1014
1015
LP = MixedIntegerLinearProgram(maximization=True)
1016
b = LP.new_variable(nonnegative=True)
1017
1018
# We define the objective
1019
LP.set_objective(sum([b[v] for v in g]))
1020
1021
# For any edge, we define a constraint
1022
for (u,v) in g.edges(labels=None):
1023
LP.add_constraint(b[u]+b[v],max=1)
1024
#LP.add_constraint(b[u]+b[v],min=1)
1025
1026
#search through all subsets of g with order >= 3
1027
#and look for *any* subgraph isomorphic to an alpha critical graph
1028
#for any match we define a constraint
1029
1030
i = 3
1031
while i <= n:
1032
SS = Subsets(Set(V),i)
1033
for S in SS:
1034
L = [g6 for g6 in alpha_critical_graph_names if Graph(g6).order() == i]
1035
#print L
1036
for g6 in L:
1037
h = Graph(g6)
1038
if g.subgraph(S).subgraph_search(h, induced=False):
1039
1040
#print S
1041
#add constraint
1042
alpha = independence_number(h)
1043
#print h.graph6_string(), alpha
1044
LP.add_constraint(sum([b[j] for j in S]), max = alpha, name = h.graph6_string())
1045
i = i + 1
1046
1047
#for c in LP.constraints():
1048
#print c
1049
1050
# The .solve() functions returns the objective value
1051
LP.solve()
1052
1053
#return LP
1054
1055
b_sol = LP.get_values(b)
1056
return b_sol, sum(b_sol.values())
1057
1058
1059
###several invariants and auxiliary functions related to the Independence Decomposition Theorem
1060
1061
#finds all vertices with weight 1 in some max weighted stable set with wieghts in {0,1,1/2}
1062
#had problem that LP solver has small numerical errors, fixed with kludgy if condition
1063
def find_stable_ones_vertices(g):
1064
F = []
1065
alpha_f = fractional_alpha(g)
1066
for v in g.vertices():
1067
gc = copy(g)
1068
gc.delete_vertices(closed_neighborhood(gc, v))
1069
alpha_f_prime = fractional_alpha(gc)
1070
if abs(alpha_f - alpha_f_prime - 1) < .01:
1071
F.append(v)
1072
return F
1073
1074
def find_max_critical_independent_set(g):
1075
S = find_stable_ones_vertices(g)
1076
H = g.subgraph(S)
1077
return H.independent_set()
1078
1079
def critical_independence_number(g):
1080
return len(find_max_critical_independent_set(g))
1081
add_to_lists(critical_independence_number, efficient_invariants, all_invariants)
1082
1083
def card_independence_irreducible_part(g):
1084
return len(find_independence_irreducible_part(g))
1085
add_to_lists(card_independence_irreducible_part, efficient_invariants, all_invariants)
1086
1087
def find_independence_irreducible_part(g):
1088
X = find_KE_part(g)
1089
SX = Set(X)
1090
Vertices = Set(g.vertices())
1091
return list(Vertices.difference(SX))
1092
1093
#returns KE part guaranteed by Independence Decomposition Theorem
1094
def find_KE_part(g):
1095
return closed_neighborhood(g, find_max_critical_independent_set(g))
1096
1097
def card_KE_part(g):
1098
return len(find_KE_part(g))
1099
add_to_lists(card_KE_part, efficient_invariants, all_invariants)
1100
1101
def find_independence_irreducible_subgraph(g):
1102
return g.subgraph(find_independence_irreducible_part(g))
1103
1104
def find_KE_subgraph(g):
1105
return g.subgraph(find_KE_part(g))
1106
1107
1108
#make invariant from property
1109
def make_invariant_from_property(property, name=None):
1110
"""
1111
This function takes a property as an argument and returns an invariant
1112
whose value is 1 if the object has the property, else 0
1113
Optionally a name for the new property can be provided as a second argument.
1114
"""
1115
def boolean_valued_invariant(g):
1116
if property(g):
1117
return 1
1118
else:
1119
return 0
1120
1121
if name is not None:
1122
boolean_valued_invariant.__name__ = name
1123
elif hasattr(property, '__name__'):
1124
boolean_valued_invariant.__name__ = '{}_value'.format(property.__name__)
1125
else:
1126
raise ValueError('Please provide a name for the new function')
1127
1128
return boolean_valued_invariant
1129
1130
# defined by R. Pepper in an unpublished paper on graph irregularity
1131
def geometric_length_of_degree_sequence(g):
1132
return sqrt(sum(d^2 for d in g.degree()))
1133
add_to_lists(geometric_length_of_degree_sequence, efficient_invariants, all_invariants)
1134
1135
# Two Stability Theta Bound
1136
# For graphs with alpha <= 2,
1137
# lovasz_theta <= 2^(2/3)*n^(1/3)
1138
# The Sandwich Theorem by Knuth p. 47
1139
def two_stability_theta_bound(g):
1140
return 2^(2/3)*g.order()^(1/3)
1141
add_to_lists(two_stability_theta_bound, efficient_invariants, all_invariants)
1142
1143
# Lovasz Theta over Root N
1144
# The Sandwich Theorem by Knuth p. 45
1145
def lovasz_theta_over_root_n(g):
1146
return g.lovasz_theta()/sqrt(g.order())
1147
add_to_lists(lovasz_theta_over_root_n, efficient_invariants, all_invariants)
1148
1149
# Theta * Theta-Complement
1150
# The Sandwich Theorem by Knuth, p. 27
1151
def theta_theta_complement(g):
1152
return g.lovasz_theta() * g.complement().lovasz_theta()
1153
add_to_lists(theta_theta_complement, efficient_invariants, all_invariants)
1154
1155
# Depth = Order - Residue
1156
# This is the number of steps it takes for Havel-Hakimi to terminate
1157
def depth(g):
1158
return g.order()-residue(g)
1159
add_to_lists(depth, efficient_invariants, all_invariants)
1160
1161
# Lovasz Theta of the complement of the given graph
1162
def lovasz_theta_complement(g):
1163
return g.complement().lovasz_theta()
1164
add_to_lists(lovasz_theta_complement, efficient_invariants, all_invariants)
1165
1166
# N over lovasz_theta_complement
1167
# This is a lower bound for lovasz theta
1168
# The Sandwich Theorem by Knuth, p. 27
1169
def n_over_lovasz_theta_complement(g):
1170
return g.order()/lovasz_theta_complement(g)
1171
add_to_lists(n_over_lovasz_theta_complement, efficient_invariants, all_invariants)
1172
1173
# The number of vertices at even distance from v and return the max over all vertices
1174
def max_even(g):
1175
from sage.graphs.distances_all_pairs import distances_all_pairs
1176
D = distances_all_pairs(g)
1177
evens_list = []
1178
for u in D:
1179
evens = 0
1180
for v in D[u]:
1181
if D[u][v] % 2 == 0:
1182
evens += 1
1183
evens_list.append(evens)
1184
return max(evens_list)
1185
add_to_lists(max_even, efficient_invariants, all_invariants)
1186
1187
# The number of vertices at even distance from v and return the min over all vertices
1188
def min_even(g):
1189
from sage.graphs.distances_all_pairs import distances_all_pairs
1190
D = distances_all_pairs(g)
1191
evens_list = []
1192
for u in D:
1193
evens = 0
1194
for v in D[u]:
1195
if D[u][v] % 2 == 0:
1196
evens += 1
1197
evens_list.append(evens)
1198
return min(evens_list)
1199
add_to_lists(min_even, efficient_invariants, all_invariants)
1200
1201
# The number of vertices at odd distance from v and return the max over all vertices
1202
def max_odd(g):
1203
from sage.graphs.distances_all_pairs import distances_all_pairs
1204
D = distances_all_pairs(g)
1205
odds_list = []
1206
for u in D:
1207
odds = 0
1208
for v in D[u]:
1209
if D[u][v] % 2 != 0:
1210
odds += 1
1211
odds_list.append(odds)
1212
return max(odds_list)
1213
add_to_lists(max_odd, efficient_invariants, all_invariants)
1214
1215
# The number of vertices at odd distance from v and return the min over all vertices
1216
def min_odd(g):
1217
from sage.graphs.distances_all_pairs import distances_all_pairs
1218
D = distances_all_pairs(g)
1219
odds_list = []
1220
for u in D:
1221
odds = 0
1222
for v in D[u]:
1223
if D[u][v] % 2 != 0:
1224
odds += 1
1225
odds_list.append(odds)
1226
return min(odds_list)
1227
add_to_lists(min_odd, efficient_invariants, all_invariants)
1228
1229
#returns sum of distances between *distinct* vertices, return infinity is graph is not connected
1230
def transmission(g):
1231
if not g.is_connected():
1232
return Infinity
1233
if g.is_tree() and max(g.degree()) == 2:
1234
summation = 0
1235
for i in [1..(g.order()-1)]:
1236
summation += (i*(i+1))/2
1237
return summation * 2
1238
else:
1239
V = g.vertices()
1240
D = g.distance_all_pairs()
1241
return sum([D[v][w] for v in V for w in V if v != w])
1242
add_to_lists(transmission, efficient_invariants, all_invariants)
1243
1244
def harmonic_index(g):
1245
sum = 0
1246
for edge in g.edges(labels = false):
1247
sum += (2 / (g.degree()[edge[0]] + g.degree()[edge[1]]))
1248
return sum
1249
add_to_lists(harmonic_index, efficient_invariants, all_invariants)
1250
1251
def bavelas_index(g):
1252
"""
1253
returns sum over all edges (v,w) of (distance from v to all other vertices)/(distance from w to all other vertices)
1254
computes each edge twice (once with v computation in numerator, once with w computation in numerator)
1255
1256
sage: bavelas_index(p6)
1257
5086/495
1258
sage: bavelas_index(k4)
1259
12
1260
"""
1261
D = g.distance_all_pairs()
1262
1263
def s_aux(v):
1264
"""
1265
computes sum of distances from v to all other vertices
1266
"""
1267
sum = 0
1268
for w in g.vertices():
1269
sum += D[v][w]
1270
return sum
1271
1272
sum_final = 0
1273
1274
for edge in g.edges(labels=false):
1275
v = edge[0]
1276
w = edge[1]
1277
sum_final += (s_aux(w) / s_aux(v)) + (s_aux(v) / s_aux(w))
1278
1279
return sum_final
1280
add_to_lists(bavelas_index, efficient_invariants, all_invariants)
1281
1282
#a solution of the invariant interpolation problem for upper bound of chromatic number for c8chords
1283
#all upper bounds in theory have value at least 3 for c8chords
1284
#returns 2 for bipartite graphs, order for non-bipartite
1285
def bipartite_chromatic(g):
1286
if g.is_bipartite():
1287
return 2
1288
else:
1289
return g.order()
1290
add_to_lists(bipartite_chromatic, efficient_invariants, all_invariants)
1291
1292
def beauchamp_index(g):
1293
"""
1294
Defined on page 597 of Sabidussi, Gert. "The centrality index of a graph." Psychometrika 31.4 (1966): 581-603.
1295
1296
sage: beauchamp_index(c4)
1297
1
1298
sage: beauchamp_index(p5)
1299
137/210
1300
sage: beauchamp_index(pete)
1301
2/3
1302
"""
1303
1304
D = g.distance_all_pairs()
1305
1306
def s_aux(v): #computes sum of distances from v to all other vertices
1307
sum = 0
1308
for w in g.vertices():
1309
sum += D[v][w]
1310
return sum
1311
1312
sum_final = 0
1313
1314
for v in g.vertices():
1315
sum_final += 1/s_aux(v)
1316
1317
print sum_final
1318
add_to_lists(beauchamp_index, efficient_invariants, all_invariants)
1319
1320
def subcubic_tr(g):
1321
"""
1322
Returns the maximum number of vertex disjoint triangles of the graph
1323
1324
Harant, Jochen, et al. "The independence number in graphs of maximum degree three." Discrete Mathematics 308.23 (2008): 5829-5833.
1325
"""
1326
return len(form_triangles_graph(g).connected_components())
1327
add_to_lists(subcubic_tr, efficient_invariants, all_invariants)
1328
1329
def edge_clustering_centrality(g, edge = None):
1330
"""
1331
Returns edge clustering centrality for all edges in a list, or a single centrality for the given edge
1332
Utility to be used with min, avg, max invariants
1333
INPUT: g - a graph
1334
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.
1335
From:
1336
An Application of Edge Clustering Centrality to Brain Connectivity by Joy Lind, Frank Garcea, Bradford Mahon, Roger Vargas, Darren A. Narayan
1337
"""
1338
if edge == None:
1339
edge_clusering_centralities = []
1340
for e in g.edges(labels = False):
1341
sum = 0
1342
for v in g.vertices():
1343
if g.subgraph(g.neighbors(v) + [v]).has_edge(e):
1344
sum += 1
1345
edge_clusering_centralities.append(sum)
1346
return edge_clusering_centralities
1347
else:
1348
for v in g.vertices():
1349
sum = 0
1350
if g.subgraph(g.neighbors(v) + [v]).has_edge(edge):
1351
sum += 1
1352
return sum
1353
1354
def max_edge_clustering_centrality(g):
1355
"""
1356
sage: max_edge_clustering_centrality(p3)
1357
2
1358
sage: max_edge_clustering_centrality(paw)
1359
3
1360
"""
1361
return max(edge_clustering_centrality(g))
1362
add_to_lists(max_edge_clustering_centrality, efficient_invariants, all_invariants)
1363
1364
def min_edge_clustering_centrality(g):
1365
"""
1366
sage: min_edge_clustering_centrality(p3)
1367
2
1368
sage: min_edge_clustering_centrality(paw)
1369
2
1370
"""
1371
return min(edge_clustering_centrality(g))
1372
add_to_lists(min_edge_clustering_centrality, efficient_invariants, all_invariants)
1373
1374
def mean_edge_clustering_centrality(g):
1375
"""
1376
sage: mean_edge_clustering_centrality(p3)
1377
2
1378
sage: mean_edge_clustering_centrality(paw)
1379
11/4
1380
"""
1381
centralities = edge_clustering_centrality(g)
1382
return sum(centralities) / len(centralities)
1383
add_to_lists(mean_edge_clustering_centrality, efficient_invariants, all_invariants)
1384
1385
def local_density(g, vertex = None):
1386
"""
1387
Returns local density for all vertices as a list, or a single local density for the given vertex
1388
INPUT: g - a graph
1389
vertex - (default: None) A vertex in g. If given, it will compute local density for just that vertex, otherwise for all of them
1390
1391
Pavlopoulos, Georgios A., et al. "Using graph theory to analyze biological networks." BioData mining 4.1 (2011): 10.
1392
"""
1393
if vertex == None:
1394
densities = []
1395
for v in g.vertices():
1396
densities.append(g.subgraph(g[v] + [v]).density())
1397
return densities
1398
return g.subgraph(g[vertex] + [vertex]).density()
1399
1400
def min_local_density(g):
1401
"""
1402
sage: min_local_density(p3)
1403
2/3
1404
sage: min_local_density(paw)
1405
2/3
1406
"""
1407
return min(local_density(g))
1408
add_to_lists(min_local_density, efficient_invariants, all_invariants)
1409
1410
def max_local_density(g):
1411
"""
1412
sage: max_local_density(p3)
1413
1
1414
sage: max_local_density(paw)
1415
1
1416
"""
1417
return max(local_density(g))
1418
add_to_lists(max_local_density, efficient_invariants, all_invariants)
1419
1420
def mean_local_density(g):
1421
"""
1422
sage: mean_local_density(p3)
1423
8/9
1424
sage: mean_local_density(paw)
1425
11/12
1426
"""
1427
densities = local_density(g)
1428
return sum(densities) / len(densities)
1429
add_to_lists(mean_local_density, efficient_invariants, all_invariants)
1430
1431
def card_simple_blocks(g):
1432
"""
1433
returns the number of blocks with order 2
1434
1435
sage: card_simple_blocks(k10)
1436
0
1437
sage: card_simple_blocks(paw)
1438
1
1439
sage: card_simple_blocks(kite_with_tail)
1440
1
1441
"""
1442
blocks = g.blocks_and_cut_vertices()[0]
1443
count = 0
1444
for block in blocks:
1445
if len(block) == 2:
1446
count += 1
1447
return count
1448
add_to_lists(card_simple_blocks, efficient_invariants, all_invariants)
1449
1450
# Block of more than 2 vertices
1451
def card_complex_blocks(g):
1452
"""
1453
returns the number of blocks with order 2
1454
1455
sage: card_complex_blocks(k10)
1456
1
1457
sage: card_complex_blocks(paw)
1458
1
1459
sage: card_complex_blocks(kite_with_tail)
1460
1
1461
"""
1462
blocks = g.blocks_and_cut_vertices()[0]
1463
count = 0
1464
for block in blocks:
1465
if len(block) > 2:
1466
count += 1
1467
return count
1468
add_to_lists(card_complex_blocks, efficient_invariants, all_invariants)
1469
1470
# Block is a clique and more than 2 vertices
1471
def card_complex_cliques(g):
1472
"""
1473
returns the number of blocks with order 2
1474
1475
sage: card_complex_clique(k10)
1476
1
1477
sage: card_complex_clique(paw)
1478
1
1479
sage: card_complex_clique(kite_with_tail)
1480
0
1481
"""
1482
blocks = g.blocks_and_cut_vertices()[0]
1483
count = 0
1484
for block in blocks:
1485
h = g.subgraph(block)
1486
if h.is_clique() and h.order() > 2:
1487
count += 1
1488
return count
1489
add_to_lists(card_complex_cliques, efficient_invariants, all_invariants)
1490
1491
def max_minus_min_degrees(g):
1492
return max_degree(g) - min_degree(g)
1493
add_to_lists(max_minus_min_degrees, efficient_invariants, all_invariants)
1494
1495
def randic_irregularity(g):
1496
return order(g)/2 - randic(g)
1497
add_to_lists(randic_irregularity, efficient_invariants, all_invariants)
1498
1499
def degree_variance(g):
1500
avg_degree = g.average_degree()
1501
return 1/order(g) * sum([d**2 - avg_degree for d in [g.degree(v) for v in g.vertices()]])
1502
add_to_lists(degree_variance, efficient_invariants, all_invariants)
1503
1504
def sum_edges_degree_difference(g):
1505
return sum([abs(g.degree(e[0]) - g.degree(e[1])) for e in g.edges()])
1506
add_to_lists(sum_edges_degree_difference, efficient_invariants, all_invariants)
1507
1508
def one_over_size_sedd(g):
1509
return 1/g.size() * sum_edges_degree_difference(g)
1510
add_to_lists(one_over_size_sedd, efficient_invariants, all_invariants)
1511
1512
def largest_eigenvalue_minus_avg_degree(g):
1513
return max_eigenvalue(g) - g.average_degree()
1514
add_to_lists(largest_eigenvalue_minus_avg_degree, efficient_invariants, all_invariants)
1515
1516
def min_betweenness_centrality(g):
1517
centralities = g.centrality_betweenness(exact=True)
1518
return centralities[min(centralities)]
1519
add_to_lists(min_betweenness_centrality, efficient_invariants, all_invariants)
1520
1521
def max_betweenness_centrality(g):
1522
centralities = g.centrality_betweenness(exact=True)
1523
return centralities[max(centralities)]
1524
add_to_lists(max_betweenness_centrality, efficient_invariants, all_invariants)
1525
1526
def mean_betweenness_centrality(g):
1527
centralities = g.centrality_betweenness(exact=True)
1528
return sum([centralities[vertex] for vertex in g.vertices()]) / g.order()
1529
add_to_lists(mean_betweenness_centrality, efficient_invariants, all_invariants)
1530
1531
def min_centrality_closeness(g):
1532
centralities = g.centrality_closeness()
1533
return centralities[min(centralities)]
1534
add_to_lists(min_centrality_closeness, efficient_invariants, all_invariants)
1535
1536
def max_centrality_closeness(g):
1537
centralities = g.centrality_closeness()
1538
return centralities[max(centralities)]
1539
add_to_lists(max_centrality_closeness, efficient_invariants, all_invariants)
1540
1541
def mean_centrality_closeness(g):
1542
centralities = g.centrality_closeness()
1543
return sum([centralities[vertex] for vertex in g.vertices()]) / g.order()
1544
add_to_lists(mean_centrality_closeness, efficient_invariants, all_invariants)
1545
1546
def min_centrality_degree(g):
1547
centralities = g.centrality_degree()
1548
return centralities[min(centralities)]
1549
add_to_lists(min_centrality_degree, efficient_invariants, all_invariants)
1550
1551
def max_centrality_degree(g):
1552
centralities = g.centrality_degree()
1553
return centralities[max(centralities)]
1554
add_to_lists(max_centrality_degree, efficient_invariants, all_invariants)
1555
1556
def mean_centrality_degree(g):
1557
centralities = g.centrality_degree()
1558
return sum([centralities[vertex] for vertex in g.vertices()]) / g.order()
1559
add_to_lists(mean_centrality_degree, efficient_invariants, all_invariants)
1560
1561
def homo_lumo_gap(g):
1562
order = g.order()
1563
if order % 2 != 0:
1564
return 0
1565
eigenvalues = g.spectrum()
1566
# Minus 1 accounts for the 0 indexing of a list
1567
return eigenvalues[floor((order+1)/2) - 1] - eigenvalues[ceil((order+1)/2) - 1]
1568
add_to_lists(homo_lumo_gap, efficient_invariants, all_invariants)
1569
1570
def homo_lumo_index(g):
1571
order = g.order()
1572
eigenvalues = g.spectrum()
1573
if order%2 == 0:
1574
# Minus 1 accounts for the 0 indexing of a list
1575
return max(abs(eigenvalues[floor((order+1)/2) - 1]), abs(eigenvalues[ceil((order+1)/2) - 1]))
1576
else:
1577
return eigenvalues[floor(order/2)]
1578
add_to_lists(homo_lumo_index, efficient_invariants, all_invariants)
1579
1580
sage_invariants = [Graph.number_of_loops, Graph.density, Graph.order, Graph.size, Graph.average_degree,
1581
Graph.triangles_count, Graph.szeged_index, Graph.radius, Graph.diameter, Graph.girth, Graph.wiener_index,
1582
Graph.average_distance, Graph.connected_components_number,
1583
Graph.maximum_average_degree, Graph.lovasz_theta, Graph.clustering_average, Graph.spanning_trees_count]
1584
1585
for i in sage_invariants:
1586
add_to_lists(i, efficient_invariants, all_invariants)
1587
1588
#####
1589
# INTRACTABLE INVATIANTS
1590
#####
1591
def domination_number(g):
1592
"""
1593
Returns the domination number of the graph g, i.e., the size of a maximum
1594
dominating set.
1595
1596
A complete graph is dominated by any of its vertices::
1597
1598
sage: domination_number(graphs.CompleteGraph(5))
1599
1
1600
1601
A star graph is dominated by its central vertex::
1602
1603
sage: domination_number(graphs.StarGraph(5))
1604
1
1605
1606
The domination number of a cycle of length n is the ceil of n/3.
1607
1608
sage: domination_number(graphs.CycleGraph(5))
1609
2
1610
"""
1611
return g.dominating_set(value_only=True)
1612
add_to_lists(domination_number, intractable_invariants, all_invariants)
1613
1614
def independence_number(g):
1615
return g.independent_set(value_only=True)
1616
add_to_lists(independence_number, intractable_invariants, all_invariants)
1617
1618
def chromatic_index(g):
1619
if g.size() == 0:
1620
return 0
1621
import sage.graphs.graph_coloring
1622
return sage.graphs.graph_coloring.edge_coloring(g, value_only=True)
1623
add_to_lists(chromatic_index, intractable_invariants, all_invariants)
1624
1625
def chromatic_num(g):
1626
return g.chromatic_number()
1627
add_to_lists(chromatic_num, intractable_invariants, all_invariants)
1628
1629
def clique_covering_number(g):
1630
# Finding the chromatic number of the complement of a fullerene
1631
# is extremely slow, even when using MILP as the algorithm.
1632
# Therefore we check to see if the graph is triangle-free.
1633
# If it is, then the clique covering number is equal to the
1634
# number of vertices minus the size of a maximum matching.
1635
if g.is_triangle_free():
1636
return g.order() - matching_number(g)
1637
gc = g.complement()
1638
return gc.chromatic_number(algorithm="MILP")
1639
add_to_lists(clique_covering_number, intractable_invariants, all_invariants)
1640
1641
def n_over_alpha(g):
1642
n = g.order() + 0.0
1643
return n/independence_number(g)
1644
add_to_lists(n_over_alpha, intractable_invariants, all_invariants)
1645
1646
def independent_dominating_set_number(g):
1647
return g.dominating_set(value_only=True, independent=True)
1648
add_to_lists(independent_dominating_set_number, intractable_invariants, all_invariants)
1649
1650
# Clearly intractable
1651
# alpha / order
1652
def independence_ratio(g):
1653
return independence_number(g)/(g.order()+0.0)
1654
add_to_lists(independence_ratio, intractable_invariants, all_invariants)
1655
1656
def min_degree_of_max_ind_set(g):
1657
"""
1658
Returns the minimum degree of any vertex that is a part of any maximum indepdendent set
1659
1660
sage: min_degree_of_vertex_in_max_ind_set(c4)
1661
2
1662
sage: min_degree_of_vertex_in_max_ind_set(pete)
1663
3
1664
"""
1665
1666
low_degree = g.order()
1667
list_of_vertices = []
1668
1669
UnionSet = Set({})
1670
IndSets = find_all_max_ind_sets(g)
1671
1672
for s in IndSets:
1673
UnionSet = UnionSet.union(Set(s))
1674
1675
list_of_vertices = list(UnionSet)
1676
1677
for v in list_of_vertices:
1678
if g.degree(v) < low_degree:
1679
low_degree = g.degree(v)
1680
1681
return low_degree
1682
add_to_lists(min_degree_of_max_ind_set, intractable_invariants, all_invariants)
1683
1684
def bipartite_number(g):
1685
"""
1686
Defined as the largest number of vertices that induces a bipartite subgraph
1687
1688
sage: bipartite_number(graphs.PetersenGraph())
1689
7
1690
sage: bipartite_number(c4)
1691
4
1692
sage: bipartite_number(graphs.CompleteGraph(3))
1693
2
1694
"""
1695
if g.is_bipartite():
1696
return g.order()
1697
return len(max_bipartite_set(g, [], g.vertices()))
1698
add_to_lists(bipartite_number, intractable_invariants, all_invariants)
1699
1700
# Needs Enhancement
1701
def edge_bipartite_number(g):
1702
"""
1703
Defined as the largest number of edges in an induced bipartite subgraph
1704
1705
sage: edge_bipartite_number(graphs.CompleteGraph(5))
1706
1
1707
sage: edge_bipartite_number(graphs.CompleteBipartiteGraph(5, 5))
1708
25
1709
sage: edge_bipartite_number(graphs.ButterflyGraph())
1710
2
1711
"""
1712
return g.subgraph(max_bipartite_set(g, [], g.vertices())).size()
1713
add_to_lists(edge_bipartite_number, intractable_invariants, all_invariants)
1714
1715
def cheeger_constant(g):
1716
"""
1717
Defined at https://en.wikipedia.org/wiki/Cheeger_constant_(graph_theory)
1718
1719
sage: cheeger_constant(p2)
1720
1
1721
sage: cheeger_constant(k5)
1722
3
1723
sage: cheeger_constant(paw)
1724
1
1725
"""
1726
n = g.order()
1727
upper = floor(n/2)
1728
1729
v = g.vertices()
1730
SetV = Set(v)
1731
1732
temp = g.order()
1733
best = n
1734
1735
for i in [1..upper]:
1736
for s in SetV.subsets(i):
1737
print 's is {}'.format(s)
1738
count = 0
1739
for u in s:
1740
print 'u is {}'.format(u)
1741
for w in SetV.difference(s):
1742
print 'w is {}'.format(w)
1743
for e in g.edges(labels=false):
1744
if Set([u,w]) == Set(e):
1745
count += 1
1746
print 'count is {}'.format(count)
1747
temp = count/i
1748
if temp < best:
1749
best = temp
1750
return best
1751
add_to_lists(cheeger_constant, intractable_invariants, all_invariants)
1752
1753
def tr(g):
1754
"""
1755
Returns the maximum number of vertex disjoint triangles of the graph
1756
1757
Harant, Jochen, et al. "The independence number in graphs of maximum degree three." Discrete Mathematics 308.23 (2008): 5829-5833.
1758
"""
1759
if is_subcubic(g):
1760
return subcubic_tr(g)
1761
return independence_number(form_triangles_graph(g))
1762
add_to_lists(tr, intractable_invariants, all_invariants)
1763
1764
def total_domination_number(g):
1765
return g.dominating_set(total=True, value_only=True)
1766
add_to_lists(total_domination_number, intractable_invariants, all_invariants)
1767
1768
add_to_lists(Graph.treewidth, intractable_invariants, all_invariants)
1769
add_to_lists(Graph.clique_number, intractable_invariants, all_invariants)
1770
1771
#FAST ENOUGH (tested for graphs on 140921): lovasz_theta, clique_covering_number, all efficiently_computable
1772
#SLOW but FIXED for SpecialGraphs
1773
1774
print("loaded invariants")
1775
1776
#############################################################################
1777
# End of invariants section #
1778
#############################################################################
1779
1780
#GRAPH PROPERTIES
1781
1782
def has_star_center(g):
1783
"""
1784
tests if graph has vertex adjacent to all others
1785
sage: has_star_center(flower_with_3_petals)
1786
True
1787
sage: has_star_center(c4)
1788
False
1789
"""
1790
n = g.order()
1791
return max_degree(g) == (n-1)
1792
1793
1794
#split graphs have the property that their complements are chordal
1795
def is_complement_of_chordal(g):
1796
"""
1797
tests is a graph is a complement of a chordal graph
1798
sage: is_complement_of_chordal(p4)
1799
True
1800
sage: is_complement_of_chordal(p5)
1801
False
1802
"""
1803
h = g.complement()
1804
return h.is_chordal()
1805
1806
#a consequence of the Friendship Theorem:
1807
#the only connected graphs where every pair of vertices has a unique neghbor are flowers
1808
def pairs_have_unique_common_neighbor(g):
1809
"""
1810
tests if a graph is a collection of disjoint triangles with a single identified vertex
1811
sage: pairs_have_unique_common_neighbor(flower(5))
1812
True
1813
sage: pairs_have_unique_common_neighbor(k3)
1814
True
1815
sage: pairs_have_unique_common_neighbor(k4)
1816
False
1817
"""
1818
if max_common_neighbors(g) != 1:
1819
return False
1820
elif min_common_neighbors(g) != 1:
1821
return False
1822
else:
1823
return True
1824
1825
#sufficient condition for hamiltonicity
1826
def is_dirac(g):
1827
n = g.order()
1828
deg = g.degree()
1829
delta=min(deg)
1830
if n/2 <= delta and n > 2:
1831
return True
1832
else:
1833
return False
1834
1835
#sufficient condition for hamiltonicity
1836
def is_ore(g):
1837
A = g.adjacency_matrix()
1838
n = g.order()
1839
D = g.degree()
1840
for i in range(n):
1841
for j in range(i):
1842
if A[i][j]==0:
1843
if D[i] + D[j] < n:
1844
return False
1845
return True
1846
1847
#sufficient condition for hamiltonicity
1848
def is_haggkvist_nicoghossian(g):
1849
k = vertex_con(g)
1850
n = g.order()
1851
delta = min(g.degree())
1852
if k >= 2 and delta >= (1.0/3)*(n+k):
1853
return True
1854
else:
1855
return False
1856
1857
#sufficient condition for hamiltonicity
1858
def is_fan(g):
1859
k = vertex_con(g)
1860
if k < 2:
1861
return False
1862
D = g.degree()
1863
Dist = g.distance_all_pairs()
1864
V = g.vertices()
1865
n = g.order()
1866
for i in range(n):
1867
for j in range (i):
1868
if Dist[V[i]][V[j]]==2 and max(D[i],D[j]) < n/2.0:
1869
return False
1870
return True
1871
1872
#sufficient condition for hamiltonicity
1873
def is_planar_transitive(g):
1874
if g.order() > 2 and g.is_planar() and g.is_vertex_transitive():
1875
return True
1876
else:
1877
return False
1878
1879
def neighbors_set(g,S):
1880
N = set()
1881
for v in S:
1882
for n in g.neighbors(v):
1883
N.add(n)
1884
return list(N)
1885
1886
#sufficient condition for hamiltonicity
1887
def is_generalized_dirac(g):
1888
n = g.order()
1889
k = vertex_con(g)
1890
if k < 2:
1891
return False
1892
for p in Subsets(g.vertices(),2):
1893
if g.is_independent_set(p):
1894
if len(neighbors_set(g,p)) < (2.0*n-1)/3:
1895
return False
1896
return True
1897
1898
#necessary condition for hamiltonicity
1899
def is_van_den_heuvel(g):
1900
n = g.order()
1901
lc = sorted(graphs.CycleGraph(n).laplacian_matrix().eigenvalues())
1902
lg = sorted(g.laplacian_matrix().eigenvalues())
1903
1904
for lci, lgi in zip(lc, lg):
1905
if lci > lgi:
1906
return False
1907
1908
def Q(g):
1909
A = g.adjacency_matrix()
1910
1911
D = A.parent(0)
1912
1913
if A.is_sparse():
1914
row_sums = {}
1915
for (i,j), entry in A.dict().iteritems():
1916
row_sums[i] = row_sums.get(i, 0) + entry
1917
for i in range(A.nrows()):
1918
D[i,i] += row_sums.get(i, 0)
1919
else:
1920
row_sums=[sum(v) for v in A.rows()]
1921
for i in range(A.nrows()):
1922
D[i,i] += row_sums[i]
1923
1924
return D + A
1925
1926
qc = sorted(Q(graphs.CycleGraph(n)).eigenvalues())
1927
qg = sorted(Q(g).eigenvalues())
1928
1929
for qci, qgi in zip(qc, qg):
1930
if qci > qgi:
1931
return False
1932
1933
return True
1934
1935
#necessary condition for hamiltonicity
1936
def is_two_connected(g):
1937
"""
1938
Returns True if the graph is 2-connected and False otherwise. A graph is
1939
2-connected if the removal of any single vertex gives a connected graph.
1940
By definition a graph on 2 or less vertices is not 2-connected.
1941
1942
sage: is_two_connected(graphs.CycleGraph(5))
1943
True
1944
sage: is_two_connected(graphs.PathGraph(5))
1945
False
1946
sage: is_two_connected(graphs.CompleteGraph(2))
1947
False
1948
sage: is_two_connected(graphs.CompleteGraph(1))
1949
False
1950
"""
1951
if g.order() <= 2:
1952
return False
1953
from itertools import combinations
1954
for s in combinations(g.vertices(), g.order() - 1):
1955
if not g.subgraph(s).is_connected():
1956
return False
1957
return True
1958
1959
#part of pebbling class0 sufficient condition
1960
def is_three_connected(g):
1961
"""
1962
Returns True if the graph is 3-connected and False otherwise. A graph is
1963
3-connected if the removal of any single vertex or any pair of vertices
1964
gives a connected graph. By definition a graph on 3 or less vertices is
1965
not 3-connected.
1966
1967
sage: is_three_connected(graphs.PetersenGraph())
1968
True
1969
sage: is_three_connected(graphs.CompleteGraph(4))
1970
True
1971
sage: is_three_connected(graphs.CycleGraph(5))
1972
False
1973
sage: is_three_connected(graphs.PathGraph(5))
1974
False
1975
sage: is_three_connected(graphs.CompleteGraph(3))
1976
False
1977
sage: is_three_connected(graphs.CompleteGraph(2))
1978
False
1979
sage: is_three_connected(graphs.CompleteGraph(1))
1980
False
1981
"""
1982
if g.order() <= 3:
1983
return False
1984
from itertools import combinations
1985
for s in combinations(g.vertices(), g.order() - 2):
1986
if not g.subgraph(s).is_connected():
1987
return False
1988
return True
1989
1990
#sufficient condition for hamiltonicity
1991
def is_lindquester(g):
1992
k = vertex_con(g)
1993
if k < 2:
1994
return False
1995
D = g.distance_all_pairs()
1996
n = g.order()
1997
V = g.vertices()
1998
for i in range(n):
1999
for j in range(i):
2000
if D[V[i]][V[j]] == 2:
2001
if len(neighbors_set(g,[V[i],V[j]])) < (2*n-1)/3.0:
2002
return False
2003
return True
2004
2005
def is_complete(g):
2006
n = g.order()
2007
e = g.size()
2008
if not g.has_multiple_edges():
2009
return e == n*(n-1)/2
2010
2011
D = g.distance_all_pairs()
2012
for i in range(n):
2013
for j in range(i):
2014
if D[V[i]][V[j]] != 1:
2015
return False
2016
return True
2017
2018
def has_c4(g):
2019
return g.subgraph_search(c4, induced=True) is not None
2020
2021
def is_c4_free(g):
2022
return not has_c4(g)
2023
2024
def has_paw(g):
2025
return g.subgraph_search(paw, induced=True) is not None
2026
2027
def is_paw_free(g):
2028
return not has_paw(g)
2029
2030
def has_dart(g):
2031
return g.subgraph_search(dart, induced=True) is not None
2032
2033
def is_dart_free(g):
2034
return not has_dart(g)
2035
2036
def is_p4_free(g):
2037
return not has_p4(g)
2038
2039
def has_p4(g):
2040
return g.subgraph_search(p4, induced=True) is not None
2041
2042
def has_kite(g):
2043
return g.subgraph_search(kite_with_tail, induced=True) is not None
2044
2045
def is_kite_free(g):
2046
return not has_kite(g)
2047
2048
def has_claw(g):
2049
return g.subgraph_search(graphs.ClawGraph(), induced=True) is not None
2050
2051
def is_claw_free(g):
2052
return not has_claw(g)
2053
2054
def has_H(g):
2055
return g.subgraph_search(double_fork, induced=True) is not None
2056
2057
def is_H_free(g):
2058
return not has_H(g)
2059
2060
def has_fork(g):
2061
return g.subgraph_search(fork, induced=True) is not None
2062
2063
def is_fork_free(g):
2064
return not has_fork(g)
2065
2066
def has_k4(g):
2067
return g.subgraph_search(alpha_critical_easy[2], induced=True) is not None
2068
2069
def is_k4_free(g):
2070
return not has_k4(g)
2071
2072
def is_biclique(g):
2073
"""
2074
a graph is a biclique if the vertices can be partitioned into 2 sets that induce cliques
2075
sage: is_biclique(p4)
2076
True
2077
sage: is_biclique(bow_tie)
2078
True
2079
"""
2080
gc = g.complement()
2081
return gc.is_bipartite()
2082
2083
def has_perfect_matching(g):
2084
n = g.order()
2085
if n%2 == 1:
2086
return False
2087
nu = g.matching(value_only=True)
2088
if 2*nu == n:
2089
return True
2090
else:
2091
return False
2092
2093
#true if radius equals diameter
2094
def has_radius_equal_diameter(g):
2095
return g.radius() == g.diameter()
2096
2097
#true if residue equals independence number
2098
def has_residue_equals_alpha(g):
2099
return residue(g) == independence_number(g)
2100
2101
def is_not_forest(g):
2102
return not g.is_forest()
2103
2104
2105
def bipartite_double_cover(g):
2106
return g.tensor_product(graphs.CompleteGraph(2))
2107
2108
def closed_neighborhood(g, verts):
2109
if isinstance(verts, list):
2110
neighborhood = []
2111
for v in verts:
2112
neighborhood += [v] + g.neighbors(v)
2113
return list(set(neighborhood))
2114
else:
2115
return [verts] + g.neighbors(verts)
2116
2117
#replaced with faster LP-solver is_independence_irreducible
2118
#has no non-empty critical independent set (<=> the only solution to the LP independence number relaxation is all 1/2's)
2119
def has_empty_KE_part(g):
2120
b = bipartite_double_cover(g)
2121
alpha = b.order() - b.matching(value_only=True)
2122
for v in g.vertices():
2123
test = b.copy()
2124
test.delete_vertices(closed_neighborhood(b,[(v,0), (v,1)]))
2125
alpha_test = test.order() - test.matching(value_only=True) + 2
2126
if alpha_test == alpha:
2127
return False
2128
return True
2129
2130
def is_bicritical(g):
2131
return has_empty_KE_part(g)
2132
2133
2134
# Vizing's Theorem: chromatic index of any graph is either Delta or Delta+1
2135
def is_class1(g):
2136
return chromatic_index(g) == max(g.degree())
2137
2138
def is_class2(g):
2139
return not(chromatic_index(g) == max(g.degree()))
2140
2141
def is_cubic(g):
2142
D = g.degree()
2143
return min(D) == 3 and max(D) == 3
2144
2145
#a property that applied to all entered hamiltonian graphs (before c60) but not the tutte graph, false for tutte graph
2146
def is_anti_tutte(g):
2147
if not g.is_connected():
2148
return False
2149
return independence_number(g) <= g.diameter() + g.girth()
2150
2151
#a property that applied to all entered hamiltonian graphs upto c6cc6 but not the tutte graph, false for tutte graph
2152
def is_anti_tutte2(g):
2153
if not g.is_connected():
2154
return False
2155
return independence_number(g) <= domination_number(g) + g.radius()- 1
2156
2157
#for any graph diam <= 2*radius. this property is true in the extremal case
2158
def diameter_equals_twice_radius(g):
2159
if not g.is_connected():
2160
return False
2161
return g.diameter() == 2*g.radius()
2162
2163
#for any graph diam >= radius. this property is true in the extremal case
2164
def diameter_equals_radius(g):
2165
if not g.is_connected():
2166
return False
2167
return g.diameter() == g.radius()
2168
2169
#almost all graphs have diameter equals 2
2170
def diameter_equals_two(g):
2171
if not g.is_connected():
2172
return False
2173
return g.diameter() == 2
2174
2175
def has_lovasz_theta_equals_alpha(g):
2176
if g.lovasz_theta() == independence_number(g):
2177
return True
2178
else:
2179
return False
2180
2181
def has_lovasz_theta_equals_cc(g):
2182
if g.lovasz_theta() == clique_covering_number(g):
2183
return True
2184
else:
2185
return False
2186
2187
#sufficient condition for hamiltonicity
2188
def is_chvatal_erdos(g):
2189
return independence_number(g) <= vertex_con(g)
2190
2191
#locally connected if the neighborhood of every vertex is connected (stronger than claw-free)
2192
def is_locally_connected(g):
2193
V = g.vertices()
2194
for v in V:
2195
N = g.neighbors(v)
2196
if len(N) > 0:
2197
GN = g.subgraph(N)
2198
if not GN.is_connected():
2199
return False
2200
return True
2201
2202
2203
#matching_covered if every edge is in a maximum matching (generalization of factor-covered which requires perfect matching)
2204
def matching_covered(g):
2205
g = g.copy()
2206
nu = matching_number(g)
2207
E = g.edges()
2208
for e in E:
2209
g.delete_edge(e)
2210
nu2 = matching_number(g)
2211
if nu != nu2:
2212
return False
2213
g.add_edge(e)
2214
return True
2215
2216
#a property that separates tutte from known hamiltonian examples, must be connected
2217
#radius(tutte) > center(tutte)
2218
def radius_greater_than_center(g):
2219
if not g.is_connected() or g.radius() <= card_center(g):
2220
return False
2221
else:
2222
return True
2223
2224
#a property that separates tutte from known hamiltonian examples, must be connected
2225
#avg_dist(tutte) > girth(tutte)
2226
def avg_distance_greater_than_girth(g):
2227
if not g.is_connected() or g.average_distance() <= g.girth():
2228
return False
2229
else:
2230
return True
2231
2232
#chromatic number equals min of known chi upper bounds
2233
def chi_equals_min_theory(g):
2234
chromatic_upper_theory = [brooks, wilf, welsh_powell, szekeres_wilf]
2235
min_theory = min([f(g) for f in chromatic_upper_theory])
2236
chi = chromatic_num(g)
2237
if min_theory == chi:
2238
return True
2239
else:
2240
return False
2241
2242
def is_heliotropic_plant(g):
2243
return (independence_number(g) == card_positive_eigenvalues(g))
2244
2245
def is_geotropic_plant(g):
2246
return (independence_number(g) == card_negative_eigenvalues(g))
2247
2248
#means has hamiltonian path, true iff g join a single vertex has hamiltonian cycle
2249
def is_traceable(g):
2250
gadd = g.join(Graph(1),labels="integers")
2251
return gadd.is_hamiltonian()
2252
2253
def has_residue_equals_two(g):
2254
return residue(g) == 2
2255
2256
#a necessary condition for being even hole free
2257
def is_chordal_or_not_perfect(g):
2258
if g.is_chordal():
2259
return true
2260
else:
2261
return not g.is_perfect()
2262
2263
def has_alpha_residue_equal_two(g):
2264
if residue(g) != 2:
2265
return false
2266
else:
2267
return independence_number(g) == 2
2268
2269
# for vizing's independence number conjecture
2270
def alpha_leq_order_over_two(g):
2271
return (2*independence_number(g) <= g.order())
2272
2273
2274
#in a luo-zhao sufficient condition for alpha <= n/2 (vizing's ind number conj)
2275
def order_leq_twice_max_degree(g):
2276
return (g.order() <= 2*max(g.degree()))
2277
2278
#not in properties list until meredith graph is computed
2279
#critical if connected, class 2, and removal of any edge decreases chromatic number
2280
def is_chromatic_index_critical(g):
2281
if not g.is_connected():
2282
return False
2283
Delta = max(g.degree())
2284
chi_e = chromatic_index(g)
2285
if chi_e != Delta + 1:
2286
return False
2287
2288
lg=g.line_graph()
2289
equiv_lines = lg.automorphism_group(return_group=False,orbits=true)
2290
equiv_lines_representatives = [orb[0] for orb in equiv_lines]
2291
2292
for e in equiv_lines_representatives:
2293
gc = copy(g)
2294
gc.delete_edge(e)
2295
chi_e_prime = chromatic_index(gc)
2296
if not chi_e_prime < chi_e:
2297
return False
2298
return True
2299
2300
#not in properties list
2301
#alpha(g-e) > alpha(g) for *every* edge g
2302
def is_alpha_critical(g):
2303
#if not g.is_connected():
2304
#return False
2305
alpha = independence_number(g)
2306
for e in g.edges():
2307
gc = copy(g)
2308
gc.delete_edge(e)
2309
alpha_prime = independence_number(gc)
2310
if alpha_prime <= alpha:
2311
return False
2312
return True
2313
2314
#graph is KE if matching number + independence number = n, test does *not* compute alpha
2315
def is_KE(g):
2316
return g.order() == len(find_KE_part(g))
2317
2318
#graph is KE if matching number + independence number = n, test comoutes alpha
2319
#def is_KE(g):
2320
# return (g.matching(value_only = True) + independence_number(g) == g.order())
2321
2322
#possibly faster version of is_KE (not currently in invariants)
2323
#def is_KE2(g):
2324
# return (independence_number(g) == critical_independence_number(g))
2325
2326
def is_independence_irreducible(g):
2327
return g.order() == card_independence_irreducible_part(g)
2328
2329
2330
def is_factor_critical(g):
2331
"""
2332
a graph is factor-critical if order is odd and removal of any vertex gives graph with perfect matching
2333
is_factor_critical(p3)
2334
False
2335
sage: is_factor_critical(c5)
2336
True
2337
"""
2338
if g.order() % 2 == 0:
2339
return False
2340
for v in g.vertices():
2341
gc = copy(g)
2342
gc.delete_vertex(v)
2343
if not has_perfect_matching(gc):
2344
return False
2345
return True
2346
2347
#returns a list of (necessarily non-adjacent) vertices that have the same neighbors as v if a pair exists or None
2348
def find_twins_of_vertex(g,v):
2349
L = []
2350
V = g.vertices()
2351
D = g.distance_all_pairs()
2352
for i in range(g.order()):
2353
w = V[i]
2354
if D[v][w] == 2 and g.neighbors(v) == g.neighbors(w):
2355
L.append(w)
2356
return L
2357
2358
def has_twin(g):
2359
t = find_twin(g)
2360
if t == None:
2361
return False
2362
else:
2363
return True
2364
2365
def is_twin_free(g):
2366
return not has_twin(g)
2367
2368
#returns twin pair (v,w) if one exists, else returns None
2369
#can't be adjacent
2370
def find_twin(g):
2371
2372
V = g.vertices()
2373
for v in V:
2374
Nv = set(g.neighbors(v))
2375
for w in V:
2376
Nw = set(g.neighbors(w))
2377
if v not in Nw and Nv == Nw:
2378
return (v,w)
2379
return None
2380
2381
def find_neighbor_twins(g):
2382
V = g.vertices()
2383
for v in V:
2384
Nv = g.neighbors(v)
2385
for w in Nv:
2386
if set(closed_neighborhood(g,v)) == set(closed_neighborhood(g,w)):
2387
return (v,w)
2388
return None
2389
2390
#given graph g and subset S, looks for any neighbor twin of any vertex in T
2391
#if result = T, then no twins, else the result is maximal, but not necessarily unique
2392
def find_neighbor_twin(g, T):
2393
gT = g.subgraph(T)
2394
for v in T:
2395
condition = False
2396
Nv = set(g.neighbors(v))
2397
#print "v = {}, Nv = {}".format(v,Nv)
2398
NvT = set(gT.neighbors(v))
2399
for w in Nv:
2400
NwT = set(g.neighbors(w)).intersection(set(T))
2401
if w not in T and NvT.issubset(NwT):
2402
twin = w
2403
T.append(w)
2404
condition = True
2405
#print "TWINS: v = {}, w = {}, sp3 = {}".format(v,w,sp3)
2406
break
2407
if condition == True:
2408
break
2409
2410
#if result = T, then no twins, else the result is maximal, but not necessarily unique
2411
def iterative_neighbor_twins(g, T):
2412
T2 = copy(T)
2413
find_neighbor_twin(g, T)
2414
while T2 != T:
2415
T2 = copy(T)
2416
find_neighbor_twin(g, T)
2417
return T
2418
2419
def is_cycle(g):
2420
return g.is_isomorphic(graphs.CycleGraph(g.order()))
2421
2422
2423
#can't compute membership in this class directly. instead testing isomorhism for 400 known class0 graphs
2424
def is_pebbling_class0(g):
2425
for hkey in class0graphs_dict:
2426
h = Graph(class0graphs_dict[hkey])
2427
if g.is_isomorphic(h):
2428
return True
2429
return False
2430
2431
def girth_greater_than_2log(g):
2432
return bool(g.girth() > 2*log(g.order(),2))
2433
2434
def szekeres_wilf_equals_chromatic_number(g):
2435
return szekeres_wilf(g) == g.chromatic_number()
2436
2437
2438
def localise(f):
2439
"""
2440
This function takes a property (i.e., a function taking only a graph as an argument) and
2441
returns the local variant of that property. The local variant is True if the property is
2442
True for the neighbourhood of each vertex and False otherwise.
2443
"""
2444
#create a local version of f
2445
def localised_function(g):
2446
return all((f(g.subgraph(g.neighbors(v))) if g.neighbors(v) else True) for v in g.vertices())
2447
2448
#we set a nice name for the new function
2449
if hasattr(f, '__name__'):
2450
if f.__name__.startswith('is_'):
2451
localised_function.__name__ = 'is_locally' + f.__name__[2:]
2452
elif f.__name__.startswith('has_'):
2453
localised_function.__name__ = 'has_locally' + f.__name__[2:]
2454
else:
2455
localised_function.__name__ = 'localised_' + f.__name__
2456
2457
return localised_function
2458
2459
is_locally_dirac = localise(is_dirac)
2460
is_locally_bipartite = localise(Graph.is_bipartite)
2461
2462
#old versioncted), can't seem to memoize that
2463
2464
def is_locally_two_connected(g):
2465
V = g.vertices()
2466
for v in V:
2467
N = g.neighbors(v)
2468
if len(N) > 0:
2469
GN = g.subgraph(N)
2470
if not is_two_connected(GN):
2471
return False
2472
return True
2473
2474
def has_equal_invariants(invar1, invar2, name=None):
2475
"""
2476
This function takes two invariants as an argument and returns the property that these invariants are equal.
2477
Optionally a name for the new function can be provided as a third argument.
2478
"""
2479
def equality_checker(g):
2480
return invar1(g) == invar2(g)
2481
2482
if name is not None:
2483
equality_checker.__name__ = name
2484
elif hasattr(invar1, '__name__') and hasattr(invar2, '__name__'):
2485
equality_checker.__name__ = 'has_{}_equals_{}'.format(invar1.__name__, invar2.__name__)
2486
else:
2487
raise ValueError('Please provide a name for the new function')
2488
2489
return equality_checker
2490
2491
def has_leq_invariants(invar1, invar2, name=None):
2492
"""
2493
This function takes two invariants as an argument and returns the property that the first invariant is
2494
less than or equal to the second invariant.
2495
Optionally a name for the new function can be provided as a third argument.
2496
"""
2497
def comparator(g):
2498
return invar1(g) <= invar2(g)
2499
2500
if name is not None:
2501
comparator.__name__ = name
2502
elif hasattr(invar1, '__name__') and hasattr(invar2, '__name__'):
2503
comparator.__name__ = 'has_{}_leq_{}'.format(invar1.__name__, invar2.__name__)
2504
else:
2505
raise ValueError('Please provide a name for the new function')
2506
2507
return comparator
2508
2509
def has_Havel_Hakimi_property(g, v):
2510
"""
2511
This function returns whether the vertex v in the graph g has the Havel-Hakimi
2512
property as defined in [1]. A vertex has the Havel-Hakimi property if it has
2513
maximum degree and the minimum degree of its neighbours is at least the maximum
2514
degree of its non-neigbours.
2515
2516
[1] Graphs with the strong Havel-Hakimi property, M. Barrus, G. Molnar, Graphs
2517
and Combinatorics, 2016, http://dx.doi.org/10.1007/s00373-015-1674-7
2518
2519
Every vertex in a regular graph has the Havel-Hakimi property::
2520
2521
sage: P = graphs.PetersenGraph()
2522
sage: for v in range(10):
2523
....: has_Havel_Hakimi_property(P,v)
2524
True
2525
True
2526
True
2527
True
2528
True
2529
True
2530
True
2531
True
2532
True
2533
True
2534
sage: has_Havel_Hakimi_property(Graph([[0,1,2,3],lambda x,y: False]),0)
2535
True
2536
sage: has_Havel_Hakimi_property(graphs.CompleteGraph(5),0)
2537
True
2538
"""
2539
if max_degree(g) > g.degree(v): return False
2540
2541
#handle the case where the graph is an independent set
2542
if len(g.neighbors(v)) == 0: return True
2543
2544
#handle the case where v is adjacent with all vertices
2545
if len(g.neighbors(v)) == len(g.vertices()) - 1: return True
2546
2547
return (min(g.degree(nv) for nv in g.neighbors(v)) >=
2548
max(g.degree(nnv) for nnv in g.vertices() if nnv != v and nnv not in g.neighbors(v)))
2549
2550
def has_strong_Havel_Hakimi_property(g):
2551
"""
2552
This function returns whether the graph g has the strong Havel-Hakimi property
2553
as defined in [1]. A graph has the strong Havel-Hakimi property if in every
2554
induced subgraph H of G, every vertex of maximum degree has the Havel-Hakimi
2555
property.
2556
2557
[1] Graphs with the strong Havel-Hakimi property, M. Barrus, G. Molnar, Graphs
2558
and Combinatorics, 2016, http://dx.doi.org/10.1007/s00373-015-1674-7
2559
2560
The graph obtained by connecting two cycles of length 3 by a single edge has
2561
the strong Havel-Hakimi property::
2562
2563
sage: has_strong_Havel_Hakimi_property(Graph('E{CW'))
2564
True
2565
"""
2566
for S in Subsets(g.vertices()):
2567
if len(S)>2:
2568
H = g.subgraph(S)
2569
Delta = max_degree(H)
2570
if any(not has_Havel_Hakimi_property(H, v) for v in S if H.degree(v) == Delta):
2571
return False
2572
return True
2573
2574
# Graph is subcubic is each vertex is at most degree 3
2575
def is_subcubic(g):
2576
return max_degree(g) == 3
2577
2578
# Max and min degree varies by at most 1
2579
def is_quasi_regular(g):
2580
if max_degree(g) - min_degree(g) < 2:
2581
return true
2582
return false
2583
2584
# g is bad if a block is isomorphic to k3, c5, k4*, c5*
2585
def is_bad(g):
2586
blocks = g.blocks_and_cut_vertices()[0]
2587
# To make a subgraph of g from the ith block
2588
for i in blocks:
2589
h = g.subgraph(i)
2590
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])
2591
if boolean == True:
2592
return True
2593
return False
2594
2595
#add all properties derived from pairs of invariants
2596
invariant_relation_properties = [has_leq_invariants(f,g) for f in all_invariants for g in all_invariants if f != g]
2597
2598
2599
efficiently_computable_properties = [Graph.is_regular, Graph.is_planar,
2600
Graph.is_forest, Graph.is_eulerian, Graph.is_connected, Graph.is_clique,
2601
Graph.is_circular_planar, Graph.is_chordal, Graph.is_bipartite,
2602
Graph.is_cartesian_product,Graph.is_distance_regular, Graph.is_even_hole_free,
2603
Graph.is_gallai_tree, Graph.is_line_graph, Graph.is_overfull, Graph.is_perfect,
2604
Graph.is_split, Graph.is_strongly_regular, Graph.is_triangle_free,
2605
Graph.is_weakly_chordal, is_dirac, is_ore, is_haggkvist_nicoghossian,
2606
is_generalized_dirac, is_van_den_heuvel, is_two_connected, is_three_connected,
2607
is_lindquester, is_claw_free, has_perfect_matching, has_radius_equal_diameter,
2608
is_not_forest, is_fan, is_cubic, diameter_equals_twice_radius,
2609
diameter_equals_radius, is_locally_connected, matching_covered, is_locally_dirac,
2610
is_locally_bipartite, is_locally_two_connected, Graph.is_interval, has_paw,
2611
is_paw_free, has_p4, is_p4_free, has_dart, is_dart_free, has_kite, is_kite_free,
2612
has_H, is_H_free, has_residue_equals_two, order_leq_twice_max_degree,
2613
alpha_leq_order_over_two, is_factor_critical, is_independence_irreducible,
2614
has_twin, is_twin_free, diameter_equals_two, girth_greater_than_2log, is_cycle,
2615
pairs_have_unique_common_neighbor, has_star_center, is_complement_of_chordal,
2616
has_c4, is_c4_free, is_subcubic, is_quasi_regular, is_bad, has_k4, is_k4_free]
2617
2618
intractable_properties = [Graph.is_hamiltonian, Graph.is_vertex_transitive,
2619
Graph.is_edge_transitive, has_residue_equals_alpha, Graph.is_odd_hole_free,
2620
Graph.is_semi_symmetric, Graph.is_line_graph, is_planar_transitive, is_class1,
2621
is_class2, is_anti_tutte, is_anti_tutte2, has_lovasz_theta_equals_cc,
2622
has_lovasz_theta_equals_alpha, is_chvatal_erdos, is_heliotropic_plant,
2623
is_geotropic_plant, is_traceable, is_chordal_or_not_perfect,
2624
has_alpha_residue_equal_two]
2625
2626
removed_properties = [is_pebbling_class0]
2627
2628
#speed notes
2629
#FAST ENOUGH (tested for graphs on 140921): is_hamiltonian, is_vertex_transitive,
2630
# is_edge_transitive, has_residue_equals_alpha, is_odd_hole_free, is_semi_symmetric,
2631
# is_line_graph, is_line_graph, is_anti_tutte, is_planar_transitive
2632
#SLOW but FIXED for SpecialGraphs: is_class1, is_class2
2633
2634
properties = efficiently_computable_properties + intractable_properties
2635
properties_plus = efficiently_computable_properties + intractable_properties + invariant_relation_properties
2636
2637
2638
invariants_from_properties = [make_invariant_from_property(property) for property in properties]
2639
invariants_plus = all_invariants + invariants_from_properties
2640
2641
# Graph.is_prime removed as faulty 9/2014
2642
# built in Graph.is_transitively_reduced removed 9/2014
2643
# is_line_graph is theoretically efficient - but Sage's implementation is not 9/2014
2644
2645
# weakly_chordal = weakly chordal, i.e., the graph and its complement have no induced cycle of length at least 5
2646
2647
2648
2649
print("loaded properties")
2650
2651
#############################################################################
2652
# End of properties section #
2653
#############################################################################
2654
# THEORY
2655
all_theorems = []
2656
2657
alpha_upper_bounds = []
2658
alpha_lower_bounds = []
2659
2660
#####
2661
# ALPHA UPPER BOUNDS
2662
#####
2663
2664
# R. Pepper. Binding independence. Ph. D. Dissertation. University of Houston. Houston, TX, 2004.
2665
alpha_annihilation_bound = annihilation_number
2666
add_to_lists(alpha_annihilation_bound, alpha_upper_bounds, all_theorems)
2667
2668
# Nemhauser, George L., and Leslie Earl Trotter. "Vertex packings: structural properties and algorithms." Mathematical Programming 8.1 (1975): 232-248.
2669
# Nemhauser, George L., and Leslie E. Trotter. "Properties of vertex packing and independence system polyhedra." Mathematical Programming 6.1 (1974): 48-61.
2670
alpha_fractional_bound = fractional_alpha
2671
add_to_lists(alpha_fractional_bound, alpha_upper_bounds, all_theorems)
2672
2673
# D. M. Cvetkovic, M. Doob, and H. Sachs. Spectra of graphs. Academic Press, New York, 1980.
2674
alpha_cvetkovic_bound = cvetkovic
2675
add_to_lists(alpha_cvetkovic_bound, alpha_upper_bounds, all_theorems)
2676
2677
# Trivial
2678
alpha_trivial_bound = Graph.order
2679
add_to_lists(alpha_trivial_bound, alpha_upper_bounds, all_theorems)
2680
2681
# Lovasz Theta
2682
alpha_lovasz_theta_bound = Graph.lovasz_theta
2683
add_to_lists(alpha_lovasz_theta_bound, alpha_upper_bounds, all_theorems)
2684
2685
# R. Pepper. Binding independence. Ph. D. Dissertation. University of Houston. Houston, TX, 2004.
2686
def alpha_kwok_bound(g):
2687
return order(g) - (size(g)/max_degree(g))
2688
add_to_lists(alpha_kwok_bound, alpha_upper_bounds, all_theorems)
2689
2690
# P. Hansen and M. Zheng. Sharp Bounds on the order, size, and stability number of graphs. NETWORKS 23 (1993), no. 2, 99-102.
2691
def alpha_hansen_bound(g):
2692
return floor(1/2 + sqrt(1/4 + order(g)**2 - order(g) - 2*size(g)))
2693
add_to_lists(alpha_hansen_bound, alpha_upper_bounds, all_theorems)
2694
2695
# Matching Number - Folklore
2696
def alpha_matching_number_bound(g):
2697
return order(g) - matching_number(g)
2698
add_to_lists(alpha_matching_number_bound, alpha_upper_bounds, all_theorems)
2699
2700
# Min-Degree Theorm
2701
def alpha_min_degree_bound(g):
2702
return order(g) - min_degree(g)
2703
add_to_lists(alpha_min_degree_bound, alpha_upper_bounds, all_theorems)
2704
2705
# Cut Vertices Theorem
2706
# Three Bounds on the Independence Number of a Graph - C. E. Larson, R. Pepper
2707
def alpha_cut_vertices_bound(g):
2708
return (g.order() - (card_cut_vertices(g)/2) - (1/2))
2709
add_to_lists(alpha_cut_vertices_bound, alpha_upper_bounds, all_theorems)
2710
2711
# Median Degree Theorem
2712
# Three Bounds on the Independence Number of a Graph - C. E. Larson, R. Pepper
2713
def alpha_median_degree_bound(g):
2714
return (g.order() - (median_degree(g)/2) - 1/2)
2715
add_to_lists(alpha_median_degree_bound, alpha_upper_bounds, all_theorems)
2716
2717
# Godsil-Newman Upper Bound theorem
2718
# Godsil, Chris D., and Mike W. Newman. "Eigenvalue bounds for independent sets." Journal of Combinatorial Theory, Series B 98.4 (2008): 721-734.
2719
def alpha_godsil_newman_bound(g):
2720
L = max(g.laplacian_matrix().change_ring(RDF).eigenvalues())
2721
return g.order()*(L-min_degree(g))/L
2722
add_to_lists(alpha_godsil_newman_bound, alpha_upper_bounds, all_theorems)
2723
2724
# AGX Upper Bound Theorem
2725
#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.
2726
def alpha_AGX_upper_bound(g):
2727
return (g.order() + max_degree(g) - ceil(2 * sqrt(g.order() - 1)))
2728
add_to_lists(alpha_AGX_upper_bound, alpha_upper_bounds, all_theorems)
2729
2730
def alpha_li_zhang_1_bound(g):
2731
"""
2732
From:
2733
A note on eigenvalue bounds for independence numbers of non-regular graphs
2734
By Yusheng Li, Zhen Zhang
2735
"""
2736
return (((max_degree(g)) - min_degree(g) - min_eigenvalue(g)) / max_degree(g)) * g.order()
2737
add_to_lists(alpha_li_zhang_1_bound, alpha_upper_bounds, all_theorems)
2738
2739
def alpha_li_zhang_2_bound(g):
2740
"""
2741
From:
2742
A note on eigenvalue bounds for independence numbers of non-regular graphs
2743
By Yusheng Li, Zhen Zhang
2744
"""
2745
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()
2746
add_to_lists(alpha_li_zhang_2_bound, alpha_upper_bounds, all_theorems)
2747
2748
def alpha_haemers_bound(g):
2749
"""
2750
From: W. Haemers, Interlacing eigenvalues and graphs, Linear Algebra Appl. 226/228 (1995) 593–616.
2751
"""
2752
return ((-max_eigenvalue(g) * min_eigenvalue(g)) / (min_degree(g)**2 - (max_eigenvalue(g) * min_eigenvalue(g)))) * g.order()
2753
add_to_lists(alpha_haemers_bound, alpha_upper_bounds, all_theorems)
2754
2755
#####
2756
# LOWER BOUNDS
2757
#####
2758
2759
# Radius Pendants Theorem
2760
# Three Bounds on the Independence Number of a Graph - C. E. Larson, R. Pepper
2761
def alpha_radius_pendants_bound(g):
2762
return (g.radius() + (card_pendants(g)/2) - 1)
2763
add_to_lists(alpha_radius_pendants_bound, alpha_lower_bounds, all_theorems)
2764
2765
# AGX Lower Bound Theorem
2766
# 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.
2767
def alpha_AGX_lower_bound(g):
2768
return ceil(2 * sqrt(g.order()))
2769
add_to_lists(alpha_AGX_lower_bound, alpha_lower_bounds, all_theorems)
2770
2771
def alpha_max_degree_minus_triangles_bound(g):
2772
return max_degree(g) - number_of_triangles(g)
2773
add_to_lists(alpha_max_degree_minus_triangles_bound, alpha_lower_bounds, all_theorems)
2774
2775
def alpha_order_brooks_bound(g):
2776
return ceil(order(x)/brooks(x))
2777
add_to_lists(alpha_order_brooks_bound, alpha_lower_bounds, all_theorems)
2778
2779
def alpha_szekeres_wilf_bound(g):
2780
return ceil(order(x)/szekeres_wilf(x))
2781
add_to_lists(alpha_szekeres_wilf_bound, alpha_lower_bounds, all_theorems)
2782
2783
def alpha_welsh_powell_bound(g):
2784
return ceil(g.order()/welsh_powell(g))
2785
add_to_lists(alpha_welsh_powell_bound, alpha_lower_bounds, all_theorems)
2786
2787
def alpha_staton_girth_bound(g):
2788
"""
2789
Hopkins, Glenn, and William Staton. "Girth and independence ratio." Can. Math. Bull. 25.2 (1982): 179-186.
2790
"""
2791
if g.girth() < 6:
2792
return 1
2793
else:
2794
d = max_degree(g)
2795
return order(g) * (2* d - 1) / (d^2 + 2 * d - 1)
2796
add_to_lists(alpha_staton_girth_bound, alpha_lower_bounds, all_theorems)
2797
2798
def alpha_staton_triangle_free_bound(g):
2799
"""
2800
Staton, William. "Some Ramsey-type numbers and the independence ratio." Transactions of the American Mathematical Society 256 (1979): 353-370.
2801
"""
2802
if g.is_triangle_free() and (max_degree(g) > 2):
2803
return (5 * g.order() ) / ((5 * max_degree(g)) - 1)
2804
return 1
2805
add_to_lists(alpha_staton_triangle_free_bound, alpha_lower_bounds, all_theorems)
2806
2807
alpha_average_distance_bound = Graph.average_distance
2808
add_to_lists(alpha_average_distance_bound, alpha_lower_bounds, all_theorems)
2809
2810
alpha_radius_bound = Graph.radius
2811
add_to_lists(alpha_radius_bound, alpha_lower_bounds, all_theorems)
2812
2813
alpha_residue_bound = residue
2814
add_to_lists(alpha_residue_bound, alpha_lower_bounds, all_theorems)
2815
2816
alpha_max_even_minus_even_horizontal_bound = max_even_minus_even_horizontal
2817
add_to_lists(alpha_max_even_minus_even_horizontal_bound, alpha_lower_bounds, all_theorems)
2818
2819
alpha_critical_independence_number_bound = critical_independence_number
2820
add_to_lists(alpha_critical_independence_number_bound, alpha_lower_bounds, all_theorems)
2821
2822
def alpha_max_degree_minus_number_of_triangles_bound(g):
2823
return max_degree(g) - number_of_triangles(g)
2824
add_to_lists(alpha_max_degree_minus_number_of_triangles_bound, alpha_lower_bounds, all_theorems)
2825
2826
def alpha_HHRS_bound(g):
2827
"""
2828
Returns 1 if max_degree > 3 or if g has k4 as a subgraph
2829
ONLY WORKS FOR CONNECTED GRAPHS becasue that is what we are focussing on, disconnected graphs just need to count the bad compenents
2830
2831
Harant, Jochen, et al. "The independence number in graphs of maximum degree three." Discrete Mathematics 308.23 (2008): 5829-5833.
2832
"""
2833
assert(g.is_connected() == true)
2834
if not is_subcubic(g):
2835
return 1
2836
if has_k4(g):
2837
return 1
2838
return (4*g.order() - g.size() - (1 if is_bad(g) else 0) - subcubic_tr(g)) / 7
2839
add_to_lists(alpha_HHRS_bound, alpha_lower_bounds, all_theorems)
2840
2841
def alpha_seklow_bound(g):
2842
"""
2843
Returns the Seklow bound from:
2844
Selkow, Stanley M. "A probabilistic lower bound on the independence number of graphs." Discrete Mathematics 132.1-3 (1994): 363-365.
2845
"""
2846
v_sum = 0
2847
for v in g.vertices():
2848
d = g.degree(v)
2849
v_sum += ((1/(d + 1)) * (1 + max(0, (d/(d + 1) - sum([(1/(g.degree(w) + 1)) for w in g.neighbors(v)])))))
2850
return v_sum
2851
add_to_lists(alpha_seklow_bound, alpha_lower_bounds, all_theorems)
2852
2853
def alpha_harant_bound(g):
2854
"""
2855
From:
2856
A lower bound on the independence number of a graph
2857
Jochen Harant
2858
"""
2859
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()]))
2860
add_to_lists(alpha_harant_bound, alpha_lower_bounds, all_theorems)
2861
2862
def alpha_harant_schiermeyer_bound(g):
2863
"""
2864
From:
2865
On the independence number of a graph in terms of order and size
2866
By J. Harant and I. Schiermeyerb
2867
"""
2868
order = g.order()
2869
t = 2*g.size() + order + 1
2870
return (t - sqrt(t**2 - 4 * order**2)) / 2
2871
add_to_lists(alpha_harant_schiermeyer_bound, alpha_lower_bounds, all_theorems)
2872
2873
def alpha_shearer_bound(g):
2874
"""
2875
From:
2876
Shearer, James B. "The independence number of dense graphs with large odd girth." Electron. J. Combin 2.2 (1995).
2877
"""
2878
girth = g.girth()
2879
2880
if girth == +Infinity:
2881
return 1.0
2882
if is_even(girth):
2883
return 1.0
2884
2885
k = ((girth - 1) / 2.0)
2886
v_sum = sum([g.degree(v)**(1/(k - 1.0)) for v in g.vertices()])
2887
return 2**(-((k - 1.0)/k)) * v_sum**((k - 1.0)/k)
2888
add_to_lists(alpha_shearer_bound, alpha_lower_bounds, all_theorems)
2889
2890
# Adds all theorems to the efficient_invariants list
2891
# Does not insert duplicates if the method is already in efficient_invariants
2892
for t in all_theorems:
2893
if t not in efficient_invariants:
2894
add_to_lists(t, efficient_invariants, theorem_invariants, all_invariants)
2895
2896
print("loaded theorems")
2897
2898
#############################################################################
2899
# End of theorems section #
2900
#############################################################################
2901
# Graph Lists
2902
2903
import pickle, os
2904
2905
graph_objects = []
2906
alpha_critical_easy = []
2907
alpha_critical_hard = []
2908
chromatic_index_critical = []
2909
chromatic_index_critical_7 = []
2910
class0graphs = []
2911
class0small = []
2912
counter_examples = []
2913
problem_graphs = []
2914
sloane_graphs = []
2915
non_connected_graphs = []
2916
dimacs_graphs = []
2917
all_graphs = []
2918
2919
# HexahedralGraph is CE to (((is_planar)&(is_regular))&(is_bipartite))->(has_residue_equals_alpha)
2920
# WagnerGraph is a graph for which the Cvetkovic bound is the best upper bound present in the Willis Thesis
2921
# OctohedralGraph is a graph for which the minimum degree is the best upper bound present in the Willis thesis
2922
# BidiakisCube is a graph where none of the upper or lower bounds in the Willis thesis give the exact value for alpha
2923
# TetrahedralGraph and MoserSpindle in the alpha critical list as "C~" and "FzEKW" respectively
2924
# MeredithGraph and SchlaefliGraph are in the Problem Graphs list
2925
2926
sage_graphs = [graphs.BullGraph(), graphs.ButterflyGraph(), graphs.ClawGraph(),
2927
graphs.DiamondGraph(), graphs.HouseGraph(), graphs.HouseXGraph(), graphs.Balaban10Cage(),
2928
graphs.Balaban11Cage(), graphs.BidiakisCube(),
2929
graphs.BiggsSmithGraph(), graphs.BlanusaFirstSnarkGraph(), graphs.BlanusaSecondSnarkGraph(),
2930
graphs.BrinkmannGraph(), graphs.BrouwerHaemersGraph(), graphs.BuckyBall(),
2931
graphs.ChvatalGraph(), graphs.ClebschGraph(),
2932
graphs.CoxeterGraph(), graphs.DesarguesGraph(), graphs.DejterGraph(), graphs.DoubleStarSnark(),
2933
graphs.DurerGraph(), graphs.DyckGraph(), graphs.EllinghamHorton54Graph(),
2934
graphs.EllinghamHorton78Graph(), graphs.ErreraGraph(), graphs.F26AGraph(), graphs.FlowerSnark(),
2935
graphs.FolkmanGraph(), graphs.FosterGraph(), graphs.FranklinGraph(), graphs.FruchtGraph(),
2936
graphs.GoldnerHararyGraph(), graphs.GossetGraph(), graphs.GrayGraph(), graphs.GrotzschGraph(),
2937
graphs.HallJankoGraph(), graphs.HarborthGraph(), graphs.HarriesGraph(), graphs.HarriesWongGraph(),
2938
graphs.HeawoodGraph(), graphs.HerschelGraph(), graphs.HigmanSimsGraph(), graphs.HoffmanGraph(),
2939
graphs.HoffmanSingletonGraph(), graphs.HoltGraph(), graphs.HortonGraph(), graphs.JankoKharaghaniTonchevGraph(), graphs.KittellGraph(),
2940
graphs.KrackhardtKiteGraph(), graphs.Klein3RegularGraph(), graphs.Klein7RegularGraph(),
2941
graphs.LjubljanaGraph(),
2942
graphs.MarkstroemGraph(), graphs.McGeeGraph(),
2943
graphs.MoebiusKantorGraph(), graphs.NauruGraph(), graphs.PappusGraph(),
2944
graphs.PoussinGraph(), graphs.PerkelGraph(), graphs.PetersenGraph(), graphs.RobertsonGraph(),
2945
graphs.ShrikhandeGraph(), graphs.SimsGewirtzGraph(),
2946
graphs.SousselierGraph(), graphs.SylvesterGraph(), graphs.SzekeresSnarkGraph(),
2947
graphs.ThomsenGraph(), graphs.TietzeGraph(),
2948
graphs.TruncatedTetrahedralGraph(), graphs.Tutte12Cage(), graphs.TutteCoxeterGraph(),
2949
graphs.TutteGraph(), graphs.WagnerGraph(), graphs.WatkinsSnarkGraph(), graphs.WellsGraph(),
2950
graphs.WienerArayaGraph(),
2951
graphs.HexahedralGraph(), graphs.DodecahedralGraph(), graphs.OctahedralGraph(), graphs.IcosahedralGraph()]
2952
2953
try:
2954
add_to_lists(graphs.CameronGraph(), sage_graphs)
2955
except Exception as e:
2956
print("The Cameron graph was not loaded. Caused by:")
2957
print(e)
2958
2959
try:
2960
add_to_lists(graphs.M22Graph(), sage_graphs)
2961
except Exception as e:
2962
print("The M22 graph was not loaded. Caused by:")
2963
print(e)
2964
2965
try:
2966
add_to_lists(graphs.TruncatedIcosidodecahedralGraph(), sage_graphs)
2967
except Exception as e:
2968
print("The truncated icosidodecahedral graph was not loaded. Caused by:")
2969
print(e)
2970
2971
try:
2972
add_to_lists(graphs.McLaughlinGraph(), sage_graphs)
2973
except Exception as e:
2974
print("The McLaughlin graph was not loaded. Caused by:")
2975
print(e)
2976
2977
try:
2978
add_to_lists(graphs.LocalMcLaughlinGraph(), sage_graphs)
2979
except Exception as e:
2980
print("The local McLaughlin graph was not loaded. Caused by:")
2981
print(e)
2982
2983
#hard built-in Sage graphs
2984
add_to_lists(graphs.IoninKharaghani765Graph(), problem_graphs, all_graphs)
2985
2986
2987
#These built in graphs are nameless so here they are given names
2988
2989
try:
2990
cell120 = graphs.Cell120()
2991
cell120.name(new = "Cell120")
2992
add_to_lists(cell120, sage_graphs)
2993
except Exception as e:
2994
print("The graph of the 120-cell was not loaded. Caused by:")
2995
print(e)
2996
2997
try:
2998
cell600 = graphs.Cell600()
2999
cell600.name(new = "Cell600")
3000
add_to_lists(cell600, sage_graphs)
3001
except Exception as e:
3002
print("The graph of the 600-cell was not loaded. Caused by:")
3003
print(e)
3004
3005
mathon_strongly_regular0 = graphs.MathonStronglyRegularGraph(0)
3006
mathon_strongly_regular0.name(new = "Mathon Strongly Regular Graph 0")
3007
3008
mathon_strongly_regular1 = graphs.MathonStronglyRegularGraph(1)
3009
mathon_strongly_regular1.name(new = "Mathon Strongly Regular Graph 1")
3010
3011
mathon_strongly_regular2 = graphs.MathonStronglyRegularGraph(2)
3012
mathon_strongly_regular2.name(new = "Mathon Strongly Regular Graph 2")
3013
3014
janko_kharaghani936 = graphs.JankoKharaghaniGraph(936)
3015
janko_kharaghani936.name(new = "Janko-Kharaghani 936")
3016
3017
janko_kharaghani1800 = graphs.JankoKharaghaniGraph(1800)
3018
janko_kharaghani1800.name(new = "Janko-Kharagani 1800")
3019
3020
for graph in sage_graphs + [mathon_strongly_regular0, mathon_strongly_regular1, mathon_strongly_regular2, janko_kharaghani936, janko_kharaghani1800]:
3021
add_to_lists(graph, graph_objects, all_graphs)
3022
3023
add_to_lists(graphs.WorldMap(), non_connected_graphs, all_graphs)
3024
3025
# Meredith graph is 4-reg, class2, non-hamiltonian: http://en.wikipedia.org/wiki/Meredith_graph
3026
add_to_lists(graphs.MeredithGraph(), problem_graphs, all_graphs)
3027
add_to_lists(graphs.SchlaefliGraph(), problem_graphs, all_graphs)
3028
3029
# A graph is alpha_critical if removing any edge increases independence number
3030
# All alpha critical graphs of orders 2 to 9, 53 in total
3031
3032
# "E|OW" is CE to (has_alpha_residue_equal_two)->((is_perfect)|(is_regular))
3033
3034
alpha_critical_graph_names = ['A_','Bw', 'C~', 'Dhc', 'D~{', 'E|OW', 'E~~w', 'FhCKG', 'F~[KG',
3035
'FzEKW', 'Fn[kG', 'F~~~w', 'GbL|TS', 'G~?mvc', 'GbMmvG', 'Gb?kTG', 'GzD{Vg', 'Gb?kR_', 'GbqlZ_',
3036
'GbilZ_', 'G~~~~{', 'GbDKPG', 'HzCGKFo', 'H~|wKF{', 'HnLk]My', 'HhcWKF_', 'HhKWKF_', 'HhCW[F_',
3037
'HxCw}V`', 'HhcGKf_', 'HhKGKf_', 'Hh[gMEO', 'HhdGKE[', 'HhcWKE[', 'HhdGKFK', 'HhCGGE@', 'Hn[gGE@',
3038
'Hn^zxU@', 'HlDKhEH', 'H~~~~~~', 'HnKmH]N', 'HnvzhEH', 'HhfJGE@', 'HhdJGM@', 'Hj~KHeF', 'HhdGHeB',
3039
'HhXg[EO', 'HhGG]ES', 'H~Gg]f{', 'H~?g]vs', 'H~@w[Vs', 'Hn_k[^o']
3040
3041
for s in alpha_critical_graph_names:
3042
g = Graph(s)
3043
g.name(new="alpha_critical_"+ s)
3044
add_to_lists(g, alpha_critical_easy, graph_objects, all_graphs)
3045
3046
# All order-7 chromatic_index_critical_graphs (and all are overfull)
3047
n7_chromatic_index_critical_names = ['FhCKG', 'FzCKW', 'FzNKW', 'FlSkG', 'Fn]kG', 'FlLKG', 'FnlkG', 'F~|{G', 'FnlLG', 'F~|\\G',
3048
'FnNLG', 'F~^LW', 'Fll\\G', 'FllNG', 'F~l^G', 'F~|^w', 'F~~^W', 'Fnl^W', 'FlNNG', 'F|\\Kg',
3049
'F~^kg', 'FlKMG']
3050
3051
for s in n7_chromatic_index_critical_names:
3052
g=Graph(s)
3053
g.name(new="chromatic_index_critical_7_" + s)
3054
add_to_lists(g, chromatic_index_critical, chromatic_index_critical_7, problem_graphs, all_graphs)
3055
3056
alpha_critical_hard = [Graph('Hj\\x{F{')]
3057
for g in alpha_critical_hard:
3058
add_to_lists(g, problem_graphs, all_graphs)
3059
3060
# Graph objects
3061
3062
p3 = graphs.PathGraph(3)
3063
p3.name(new = "p3")
3064
add_to_lists(p3, graph_objects, all_graphs)
3065
3066
p4 = graphs.PathGraph(4)
3067
p4.name(new="p4")
3068
add_to_lists(p4, graph_objects, all_graphs)
3069
3070
p5 = graphs.PathGraph(5)
3071
p5.name(new = "p5")
3072
add_to_lists(p5, graph_objects, all_graphs)
3073
3074
p6 = graphs.PathGraph(6)
3075
p6.name(new="p6")
3076
add_to_lists(p6, graph_objects, all_graphs)
3077
3078
"""
3079
CE to independence_number(x) <= e^(cosh(max_degree(x) - 1))
3080
and to
3081
independence_number(x) <= max_degree(x)*min_degree(x) + card_periphery(x)
3082
"""
3083
p9 = graphs.PathGraph(9)
3084
p9.name(new = "p9")
3085
add_to_lists(p9, graph_objects, counter_examples, all_graphs)
3086
3087
"""
3088
P29 is a CE to independence_number(x) <=degree_sum(x)/sqrt(card_negative_eigenvalues(x))
3089
and to
3090
<= max_degree(x)^e^card_center(x)
3091
and to
3092
<= max_degree(x)^2 + card_periphery(x)
3093
"""
3094
p29 = graphs.PathGraph(29)
3095
p29.name(new = "p29")
3096
add_to_lists(p29, graph_objects, counter_examples, all_graphs)
3097
3098
# CE to independence_number(x) <= 2*cvetkovic(x)*log(10)/log(x.size())
3099
p102 = graphs.PathGraph(102)
3100
p102.name(new = "p102")
3101
add_to_lists(p102, graph_objects, counter_examples, all_graphs)
3102
3103
# CE to independence_number(x)<=welsh_powell(x)^e^different_degrees(x)
3104
p6707 = graphs.PathGraph(6707)
3105
p6707.name(new = "p6707")
3106
3107
c4 = graphs.CycleGraph(4)
3108
c4.name(new="c4")
3109
add_to_lists(c4, graph_objects, all_graphs)
3110
3111
c6 = graphs.CycleGraph(6)
3112
c6.name(new = "c6")
3113
add_to_lists(c6, graph_objects, all_graphs)
3114
3115
# CE to independence_number(x) <= (e^welsh_powell(x) - graph_rank(x))^2
3116
c22 = graphs.CycleGraph(22)
3117
c22.name(new = "c22")
3118
add_to_lists(c22, graph_objects, counter_examples, all_graphs)
3119
3120
# CE to independence_number(x) <= minimum(cvetkovic(x), 2*e^sum_temperatures(x))
3121
c34 = graphs.CycleGraph(34)
3122
c34.name(new = "c34")
3123
add_to_lists(c34, graph_objects, counter_examples, all_graphs)
3124
3125
# CE to independence_number(x) <= residue(x)^(degree_sum(x)^density(x))
3126
c102 = graphs.CycleGraph(102)
3127
c102.name(new = "c102")
3128
add_to_lists(c102, graph_objects, counter_examples, all_graphs)
3129
3130
"""
3131
Sage defines circulant graphs without the cycle, whereas the paper defines it with the cycle
3132
From:
3133
Brimkov, Valentin. "Algorithmic and explicit determination of the Lovasz number for certain circulant graphs." Discrete Applied Mathematics 155.14 (2007): 1812-1825.
3134
"""
3135
c13_2 = graphs.CirculantGraph(13, 2)
3136
c13_2.add_cycle([0..12])
3137
c13_2.name(new = "c13_2")
3138
add_to_lists(c13_2, graph_objects, all_graphs)
3139
3140
k3 = alpha_critical_easy[1]
3141
k4 = alpha_critical_easy[2]
3142
3143
k10 = graphs.CompleteGraph(10)
3144
k10.name(new="k10")
3145
add_to_lists(k10, graph_objects, all_graphs)
3146
3147
# CE to independence_number(x) >= floor(tan(floor(gutman_energy(x))))
3148
k37 = graphs.CompleteGraph(37)
3149
k37.name(new = "k37")
3150
add_to_lists(k37, graph_objects, counter_examples, all_graphs)
3151
3152
# Lipták, László, and László Lovász. "Critical facets of the stable set polytope." Combinatorica 21.1 (2001): 61-88.
3153
k1_4 = graphs.CompleteBipartiteGraph(1, 4)
3154
k1_4.name(new = "k1_4")
3155
add_to_lists(k1_4, graph_objects, all_graphs)
3156
3157
# A 4 ray star with each ray subdivided into two edges
3158
# Lipták, László, and László Lovász. "Critical facets of the stable set polytope." Combinatorica 21.1 (2001): 61-88.
3159
k1_4_sub2 = Graph("H?EA@?^")
3160
k1_4_sub2.name(new = "k1_4_sub2")
3161
add_to_lists(k1_4_sub2, graph_objects, all_graphs)
3162
3163
# CE to independence_number(x) <= minimum(lovasz_theta(x), 2*e^sum_temperatures(x))
3164
# and to
3165
# independence_number(x) <= minimum(floor(lovasz_theta(x)), 2*e^sum_temperatures(x))
3166
# and to
3167
# independence_number(x) >= -brinkmann_steffen(x) + 1/2*card_center(x)
3168
k1_9 = graphs.CompleteBipartiteGraph(1,9)
3169
k1_9.name(new = "k1_9")
3170
add_to_lists(k1_9, graph_objects, counter_examples, all_graphs)
3171
3172
# The line graph of k3,3
3173
k3_3_line_graph = graphs.CompleteBipartiteGraph(3, 3).line_graph()
3174
k3_3_line_graph.name(new = "k3_3 line graph")
3175
add_to_lists(k3_3_line_graph, graph_objects, all_graphs)
3176
3177
k5_3=graphs.CompleteBipartiteGraph(5,3)
3178
k5_3.name(new = "k5_3")
3179
add_to_lists(k5_3, graph_objects, all_graphs)
3180
3181
# CE to independence_number(x) <= diameter^(max_degree-1)
3182
# diameter is 16, Delta=3, alpha = 341
3183
bt2_8 = graphs.BalancedTree(2,8)
3184
bt2_8.name(new = "bt2_8")
3185
add_to_lists(bt2_8, graph_objects, counter_examples, all_graphs)
3186
3187
#two c4's joined at a vertex
3188
c4c4=graphs.CycleGraph(4)
3189
for i in [4,5,6]:
3190
c4c4.add_vertex()
3191
c4c4.add_edge(3,4)
3192
c4c4.add_edge(5,4)
3193
c4c4.add_edge(5,6)
3194
c4c4.add_edge(6,3)
3195
c4c4.name(new="c4c4")
3196
add_to_lists(c4c4, graph_objects, all_graphs)
3197
3198
#two c5's joined at a vertex: eulerian, not perfect, not hamiltonian
3199
c5c5=graphs.CycleGraph(5)
3200
for i in [5,6,7,8]:
3201
c5c5.add_vertex()
3202
c5c5.add_edge(0,5)
3203
c5c5.add_edge(0,8)
3204
c5c5.add_edge(6,5)
3205
c5c5.add_edge(6,7)
3206
c5c5.add_edge(7,8)
3207
c5c5.name(new="c5c5")
3208
add_to_lists(c5c5, graph_objects, all_graphs)
3209
3210
K4a=graphs.CompleteGraph(4)
3211
K4b=graphs.CompleteGraph(4)
3212
K4a.delete_edge(0,1)
3213
K4b.delete_edge(0,1)
3214
regular_non_trans = K4a.disjoint_union(K4b)
3215
regular_non_trans.add_edge((0,0),(1,1))
3216
regular_non_trans.add_edge((0,1),(1,0))
3217
regular_non_trans.name(new="regular_non_trans")
3218
add_to_lists(regular_non_trans, graph_objects, all_graphs)
3219
3220
c6ee = graphs.CycleGraph(6)
3221
c6ee.add_edges([(1,5), (2,4)])
3222
c6ee.name(new="c6ee")
3223
add_to_lists(c6ee, graph_objects, all_graphs)
3224
3225
#c6ee plus another chord: hamiltonian, regular, vertex transitive
3226
c6eee = copy(c6ee)
3227
c6eee.add_edge(0,3)
3228
c6eee.name(new="c6eee")
3229
add_to_lists(c6eee, graph_objects, all_graphs)
3230
3231
#c8 plus one long vertical chord and 3 parallel horizontal chords
3232
c8chorded = graphs.CycleGraph(8)
3233
c8chorded.add_edge(0,4)
3234
c8chorded.add_edge(1,7)
3235
c8chorded.add_edge(2,6)
3236
c8chorded.add_edge(3,5)
3237
c8chorded.name(new="c8chorded")
3238
add_to_lists(c8chorded, graph_objects, all_graphs)
3239
3240
#c8 plus 2 parallel chords: hamiltonian, tri-free, not vertex-transitive
3241
c8chords = graphs.CycleGraph(8)
3242
c8chords.add_edge(1,6)
3243
c8chords.add_edge(2,5)
3244
c8chords.name(new="c8chords")
3245
add_to_lists(c8chords, graph_objects, all_graphs)
3246
3247
# C10 with an edge through the center
3248
c10e = graphs.CycleGraph(10)
3249
c10e.add_edge(0,5)
3250
c10e.name(new = "c10e")
3251
add_to_lists(c10e, graph_objects, all_graphs)
3252
3253
# C10e with another edge through center
3254
c10ee = graphs.CycleGraph(10)
3255
c10ee.add_edges([(0,5), (9, 4)])
3256
c10ee.name(new = "c10ee")
3257
add_to_lists(c10ee, graph_objects, all_graphs)
3258
3259
prismsub = graphs.CycleGraph(6)
3260
prismsub.add_edge(0,2)
3261
prismsub.add_edge(3,5)
3262
prismsub.add_edge(1,4)
3263
prismsub.subdivide_edge(1,4,1)
3264
prismsub.name(new="prismsub")
3265
add_to_lists(prismsub, graph_objects, all_graphs)
3266
3267
# ham, not vertex trans, tri-free, not cartesian product
3268
prismy = graphs.CycleGraph(8)
3269
prismy.add_edge(2,5)
3270
prismy.add_edge(0,3)
3271
prismy.add_edge(4,7)
3272
prismy.name(new="prismy")
3273
add_to_lists(prismy, graph_objects, all_graphs)
3274
3275
#c10 with chords, ham, tri-free, regular, planar, vertex transitive
3276
sixfour = graphs.CycleGraph(10)
3277
sixfour.add_edge(1,9)
3278
sixfour.add_edge(0,2)
3279
sixfour.add_edge(3,8)
3280
sixfour.add_edge(4,6)
3281
sixfour.add_edge(5,7)
3282
sixfour.name(new="sixfour")
3283
add_to_lists(sixfour, graph_objects, all_graphs)
3284
3285
#unique 24-vertex fullerene: hamiltonian, planar, not vertex transitive
3286
c24 = Graph('WsP@H?PC?O`?@@?_?GG@??CC?G??GG?E???o??B???E???F')
3287
c24.name(new="c24")
3288
add_to_lists(c24, graph_objects, all_graphs)
3289
3290
#unique 26-atom fullerene: hamiltonian, planar, not vertex trans, radius=5, diam=6
3291
c26 = Graph('YsP@H?PC?O`?@@?_?G?@??CC?G??GG?E??@_??K???W???W???H???E_')
3292
c26.name(new="c26")
3293
add_to_lists(c26, graph_objects, all_graphs)
3294
3295
#the unique 100-atom fullerene with minimum independence number of 43 (and IPR, tetrahedral symmetry)
3296
c100 = 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")
3297
c100.name(new="c100")
3298
add_to_lists(c100, graph_objects, all_graphs)
3299
3300
"""
3301
The Holton-McKay graph is the smallest planar cubic hamiltonian graph with an edge
3302
that is not contained in a hamiltonian cycle. It has 24 vertices and the edges (0,3)
3303
and (4,7) are not contained in a hamiltonian cycle. This graph was mentioned in
3304
D. A. Holton and B. D. McKay, Cycles in 3-connected cubic planar graphs II, Ars
3305
Combinatoria, 21A (1986) 107-114.
3306
3307
sage: holton_mckay
3308
holton_mckay: Graph on 24 vertices
3309
sage: holton_mckay.is_planar()
3310
True
3311
sage: holton_mckay.is_regular()
3312
True
3313
sage: max(holton_mckay.degree())
3314
3
3315
sage: holton_mckay.is_hamiltonian()
3316
True
3317
sage: holton_mckay.radius()
3318
4
3319
sage: holton_mckay.diameter()
3320
6
3321
"""
3322
holton_mckay = Graph('WlCGKS??G?_D????_?g?DOa?C?O??G?CC?`?G??_?_?_??L')
3323
holton_mckay.name(new="holton_mckay")
3324
add_to_lists(holton_mckay, graph_objects, all_graphs)
3325
3326
#an example of a bipartite, 1-tough, not van_den_heuvel, not hamiltonian graph
3327
kratsch_lehel_muller = graphs.PathGraph(12)
3328
kratsch_lehel_muller.add_edge(0,5)
3329
kratsch_lehel_muller.add_edge(6,11)
3330
kratsch_lehel_muller.add_edge(4,9)
3331
kratsch_lehel_muller.add_edge(1,10)
3332
kratsch_lehel_muller.add_edge(2,7)
3333
kratsch_lehel_muller.name(new="kratsch_lehel_muller")
3334
add_to_lists(kratsch_lehel_muller, graph_objects, all_graphs)
3335
3336
#ham, not planar, not anti_tutte
3337
c6xc6 = graphs.CycleGraph(6).cartesian_product(graphs.CycleGraph(6))
3338
c6xc6.name(new="c6xc6")
3339
add_to_lists(c6xc6, graph_objects, all_graphs)
3340
3341
c7xc7 = graphs.CycleGraph(7).cartesian_product(graphs.CycleGraph(7))
3342
c7xc7.name(new="c7xc7")
3343
add_to_lists(c7xc7, graph_objects, all_graphs)
3344
3345
# Product Graphs, fig. 1.13
3346
c6xk2 = graphs.CycleGraph(6).cartesian_product(graphs.CompleteGraph(2))
3347
c6xk2.name(new = "c6xk2")
3348
add_to_lists(c6xk2, graph_objects, all_graphs)
3349
3350
# Product Graphs, fig. 1.13
3351
k1_4xp3 = graphs.CompleteBipartiteGraph(1, 4).cartesian_product(graphs.PathGraph(3))
3352
k1_4xp3.name(new = "k1_4xp3")
3353
add_to_lists(k1_4xp3, graph_objects, all_graphs)
3354
3355
# Product Graphs, fig. 1.14
3356
p4xk3xk2 = graphs.PathGraph(4).cartesian_product(graphs.CompleteGraph(3)).cartesian_product(graphs.CompleteGraph(2))
3357
p4xk3xk2.name(new = "p4xk3xk2")
3358
add_to_lists(p4xk3xk2, graph_objects, all_graphs)
3359
3360
# Product Graphs, fig. 4.1
3361
p3xk2xk2 = graphs.PathGraph(3).cartesian_product(graphs.CompleteGraph(2)).cartesian_product(graphs.CompleteGraph(2))
3362
p3xk2xk2.name(new = "p3xk2xk2")
3363
add_to_lists(p3xk2xk2, graph_objects, all_graphs)
3364
3365
# Product Graphs, fig. 5.1
3366
p4Xp5 = graphs.PathGraph(4).strong_product(graphs.PathGraph(5))
3367
p4Xp5.name(new = "p4Xp5")
3368
add_to_lists(p4Xp5, graph_objects, all_graphs)
3369
3370
# Product Graphs, fig. 5.4
3371
p5txp3 = graphs.PathGraph(5).tensor_product(graphs.PathGraph(3))
3372
p5txp3.name(new = "p5txp3")
3373
add_to_lists(p5txp3, non_connected_graphs, all_graphs)
3374
3375
# Product Graphs, Fig 6.1
3376
k3lxp3 = graphs.CompleteGraph(3).lexicographic_product(graphs.PathGraph(3))
3377
k3lxp3.name(new = "k3lxp3")
3378
add_to_lists(k3lxp3, graph_objects, all_graphs)
3379
3380
# Product Graphs, Fig 6.1
3381
p3lxk3 = graphs.PathGraph(3).lexicographic_product(graphs.CompleteGraph(3))
3382
p3lxk3.name(new = "p3lxk3")
3383
add_to_lists(p3lxk3, graph_objects, all_graphs)
3384
3385
"""
3386
Referenced p.15
3387
Mathew, K. Ashik, and Patric RJ Östergård. "New lower bounds for the Shannon capacity of odd cycles." Designs, Codes and Cryptography (2015): 1-10.
3388
"""
3389
c5Xc5 = graphs.CycleGraph(5).strong_product(graphs.CycleGraph(5))
3390
c5Xc5.name(new = "c5Xc5")
3391
add_to_lists(c5Xc5, graph_objects, all_graphs)
3392
3393
#non-ham, 2-connected, eulerian (4-regular)
3394
gould = Graph('S~dg?CB?wC_L????_?W?F??c?@gOOOGGK')
3395
gould.name(new="gould")
3396
add_to_lists(gould, graph_objects, all_graphs)
3397
3398
#two k5s with single edge removed from each and lines joining these 4 points to a new center point, non-hamiltonian
3399
throwing = Graph('J~wWGGB?wF_')
3400
throwing.name(new="throwing")
3401
add_to_lists(throwing, graph_objects, all_graphs)
3402
3403
#k4 plus k2 on one side, open k5 on other, meet at single point in center, non-hamiltonian
3404
throwing2 = Graph("K~wWGKA?gB_N")
3405
throwing2.name(new="throwing2")
3406
add_to_lists(throwing2, graph_objects, all_graphs)
3407
3408
#similar to throwing2 with pair of edges swapped, non-hamiltonian
3409
throwing3 = Graph("K~wWGGB?oD_N")
3410
throwing3.name(new="throwing3")
3411
add_to_lists(throwing3, graph_objects, all_graphs)
3412
3413
#graph has diameter != radius but is hamiltonian
3414
tent = graphs.CycleGraph(4).join(Graph(1),labels="integers")
3415
tent.name(new="tent")
3416
add_to_lists(tent, graph_objects, all_graphs)
3417
3418
# C5 with chords from one vertex to other 2 (showed up in auto search for CE's): hamiltonian
3419
bridge = Graph("DU{")
3420
bridge.name(new="bridge")
3421
add_to_lists(bridge, graph_objects, all_graphs)
3422
3423
# nico found the smallest hamiltonian overfull graph
3424
non_ham_over = Graph("HCQRRQo")
3425
non_ham_over.name(new="non_ham_over")
3426
add_to_lists(non_ham_over, graph_objects, all_graphs)
3427
3428
# From Ryan Pepper
3429
ryan = Graph("WxEW?CB?I?_R????_?W?@?OC?AW???O?C??B???G?A?_??R")
3430
ryan.name(new="ryan")
3431
add_to_lists(ryan, graph_objects, all_graphs)
3432
3433
# Ryan Pepper
3434
# CE to independence_number(x) <= 2 * chromatic_number(x) + 2 * residue(x)
3435
# has alpha=25,chi=2,residue=10
3436
ryan2=graphs.CirculantGraph(50,[1,3])
3437
ryan2.name(new="circulant_50_1_3")
3438
add_to_lists(ryan2, graph_objects, counter_examples, all_graphs)
3439
3440
# From Ryan Pepper
3441
# CE to independence_number(x) >= diameter(x) - 1 for regular graphs
3442
pepper_1_gadget = Graph('Ot???CA?WB`_B@O_B_B?A')
3443
pepper_1_gadget.name(new = "pepper_1_gadget")
3444
add_to_lists(pepper_1_gadget, graph_objects, counter_examples, all_graphs)
3445
3446
# p10 joined to 2 points of k4
3447
# CE to chromatic_number <= avg_degree + 1
3448
p10k4=Graph('MhCGGC@?G?_@_B?B_')
3449
p10k4.name(new="p10k4")
3450
add_to_lists(p10k4, graph_objects, counter_examples, all_graphs)
3451
3452
# star on 13 points with added edge:
3453
# CE to independence_number(x) <= dom + girth(x)^2
3454
s13e = Graph('M{aCCA?_C?O?_?_??')
3455
s13e.name(new="s13e")
3456
add_to_lists(s13e, graph_objects, counter_examples, all_graphs)
3457
3458
# CE to independence_number(x) <= 2 * girth(x)^2 + 2
3459
# Star with 22 rays plus extra edge
3460
s22e = graphs.StarGraph(22)
3461
s22e.add_edge(1,2)
3462
s22e.name(new="s22e")
3463
add_to_lists(s22e, graph_objects, counter_examples, all_graphs)
3464
3465
# Graph from delavina's jets paper
3466
starfish = Graph('N~~eeQoiCoM?Y?U?F??')
3467
starfish.name(new="starfish")
3468
add_to_lists(starfish, graph_objects, all_graphs)
3469
3470
# difficult graph from INP: order=11, alpha=4, best lower bound < 3
3471
difficult11 = Graph('J?`FBo{fdb?')
3472
difficult11.name(new="difficult11")
3473
add_to_lists(difficult11, graph_objects, all_graphs)
3474
3475
# c4 joined to K# at point: not KE, alpha=theta=nu=3, delting any vertex gives KE graph
3476
c5k3=Graph('FheCG')
3477
c5k3.name(new="c5k3")
3478
add_to_lists(c5k3, graph_objects, all_graphs)
3479
3480
# mycielskian of a triangle:
3481
# CE to chi <= max(clique, nu)
3482
# chi=4, nu = clique = 3
3483
c3mycielski = Graph('FJnV?')
3484
c3mycielski.name(new="c3mycieski")
3485
add_to_lists(c3mycielski, problem_graphs , counter_examples, all_graphs)
3486
3487
# 4th mycielskian of a triangle,
3488
# CE to chi <= clique + girth
3489
# chi = 7, clique = girth = 3
3490
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????????|_????????}_????????^_?????????')
3491
c3mycielski4.name(new="c3mycielski4")
3492
add_to_lists(c3mycielski4, graph_objects, counter_examples, all_graphs)
3493
3494
# A PAW is a traingle with a pendant
3495
# Shows up in a sufficient condition for hamiltonicity
3496
paw=Graph('C{')
3497
paw.name(new="paw")
3498
add_to_lists(paw, graph_objects, all_graphs)
3499
3500
# 2 octahedrons, remove one edge from each, add vertex, connect it to deleted edge vertices
3501
# its regular of degree 4
3502
binary_octahedron = Graph('L]lw??B?oD_Noo')
3503
binary_octahedron.name(new = "binary_octahedron")
3504
add_to_lists(binary_octahedron, graph_objects, all_graphs)
3505
3506
# this graph shows that the cartesian product of 2 KE graphs is not necessarily KE
3507
# 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.
3508
paw_x_paw = paw.cartesian_product(paw)
3509
paw_x_paw.name(new = "paw_x_paw")
3510
add_to_lists(paw_x_paw, graph_objects, all_graphs)
3511
3512
#a DART is a kite with a pendant
3513
dart = Graph('DnC')
3514
dart.name(new="dart")
3515
add_to_lists(dart, graph_objects, all_graphs)
3516
3517
# CE to ((is_chordal)^(is_forest))->(has_residue_equals_alpha)
3518
ce2=Graph("HdGkCA?")
3519
ce2.name(new = "ce2")
3520
add_to_lists(ce2, graph_objects, counter_examples, all_graphs)
3521
3522
# CE to ((~(is_planar))&(is_chordal))->(has_residue_equals_alpha)
3523
ce4=Graph("G~sNp?")
3524
ce4.name(new = "ce4")
3525
add_to_lists(ce4, graph_objects, counter_examples, all_graphs)
3526
3527
# CE to (((is_line_graph)&(is_cartesian_product))|(is_split))->(has_residue_equals_alpha)
3528
ce5=Graph("X~}AHKVB{GGPGRCJ`B{GOO`C`AW`AwO`}CGOO`AHACHaCGVACG^")
3529
ce5.name(new = "ce5")
3530
add_to_lists(ce5, graph_objects, counter_examples, all_graphs)
3531
3532
# CE to (is_split)->((order_leq_twice_max_degree)&(is_chordal))
3533
ce6 = Graph("H??E@cN")
3534
ce6.name(new = "ce6")
3535
add_to_lists(ce6, graph_objects, counter_examples, all_graphs)
3536
3537
# CE to (has_residue_equals_alpha)->((is_bipartite)->(order_leq_twice_max_degree))
3538
ce7 = Graph("FpGK?")
3539
ce7.name(new = "ce7")
3540
add_to_lists(ce7, graph_objects, counter_examples, all_graphs)
3541
3542
# CE to ((has_paw)&(is_circular_planar))->(has_residue_equals_alpha)
3543
ce8 = Graph('IxCGGC@_G')
3544
ce8.name(new = "ce8")
3545
add_to_lists(ce8, graph_objects, counter_examples, all_graphs)
3546
3547
# CE to ((has_H)&(is_forest))->(has_residue_equals_alpha)
3548
ce9 = Graph('IhCGGD?G?')
3549
ce9.name(new = "ce9")
3550
add_to_lists(ce9, graph_objects, counter_examples, all_graphs)
3551
3552
# CE to (((is_eulerian)&(is_planar))&(has_paw))->(has_residue_equals_alpha)
3553
ce10=Graph('KxkGGC@?G?o@')
3554
ce10.name(new = "ce10")
3555
add_to_lists(ce10, graph_objects, counter_examples, all_graphs)
3556
3557
# CE to (((is_cubic)&(is_triangle_free))&(is_H_free))->(has_residue_equals_two)
3558
ce12 = Graph("Edo_")
3559
ce12.name(new = "ce12")
3560
add_to_lists(ce12, graph_objects, counter_examples, all_graphs)
3561
3562
# CE to ((diameter_equals_twice_radius)&(is_claw_free))->(has_residue_equals_two)
3563
ce13 = Graph("ExOG")
3564
ce13.name(new = "ce13")
3565
add_to_lists(ce13, graph_objects, counter_examples, all_graphs)
3566
3567
# CE to (~(matching_covered))->(has_residue_equals_alpha)
3568
ce14 = Graph('IhCGGC_@?')
3569
ce14.name(new = "IhCGGC_@?")
3570
add_to_lists(ce14, graph_objects, counter_examples, all_graphs)
3571
3572
"""
3573
CE to independence_number(x) <= 10^order_automorphism_group(x)
3574
3575
sage: order(ce15)
3576
57
3577
sage: independence_number(ce15)
3578
25
3579
"""
3580
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@`??")
3581
ce15.name(new = "ce15")
3582
add_to_lists(ce15, graph_objects, counter_examples, all_graphs)
3583
3584
# CE to independence_number(x) <= 2*maximum(welsh_powell(x), max_even_minus_even_horizontal(x))
3585
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?")
3586
ce16.name(new = "ce16")
3587
add_to_lists(ce16, graph_objects, counter_examples, all_graphs)
3588
3589
# CE to independence_number(x) >= 1/2*cvetkovic(x)
3590
ce17 = Graph("S??wG@@h_GWC?AHG?_gMGY_FaIOk@?C?S")
3591
ce17.name(new = "ce17")
3592
add_to_lists(ce17, graph_objects, counter_examples, all_graphs)
3593
3594
# CE to independence_number(x) >= matching_number - sigma_2
3595
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?")
3596
ce18.name(new = "ce18")
3597
add_to_lists(ce18, graph_objects, counter_examples, all_graphs)
3598
3599
# CE to independence_number(x) <= maximum(max_even_minus_even_horizontal(x), radius(x)*welsh_powell(x))
3600
ce19 = Graph('J?@OOGCgO{_')
3601
ce19.name(new = "ce19")
3602
add_to_lists(ce19, graph_objects, counter_examples, all_graphs)
3603
3604
# CE to independence_number(x) <= card_center(x) + max_even_minus_even_horizontal(x) + 1
3605
ce20 = Graph('M?CO?k?OWEQO_O]c_')
3606
ce20.name(new = "ce20")
3607
add_to_lists(ce20, graph_objects, counter_examples, all_graphs)
3608
3609
# CE to independence_number(x) <= median_degree(x)^2 + card_periphery(x)
3610
ce21 = Graph('FiQ?_')
3611
ce21.name(new = "ce21")
3612
add_to_lists(ce21, graph_objects, counter_examples, all_graphs)
3613
3614
# CE to independence_number(x) <= brinkmann_steffen(x) + max_even_minus_even_horizontal(x) + 1
3615
ce22 = Graph('Ss?fB_DYUg?gokTEAHC@ECSMQI?OO?GD?')
3616
ce22.name(new = "ce22")
3617
add_to_lists(ce22, graph_objects, counter_examples, all_graphs)
3618
3619
# CE to independence_number(x) <= inverse_degree(x) + order_automorphism_group(x) + 1
3620
ce23 = Graph("HkIU|eA")
3621
ce23.name(new = "ce23")
3622
add_to_lists(ce23, graph_objects, counter_examples, all_graphs)
3623
3624
# CE to independence_number(x) <= ceil(eulerian_faces(x)/diameter(x)) +max_even_minus_even_horizontal(x)
3625
ce24 = Graph('JCbcA?@@AG?')
3626
ce24.name(new = "ce24")
3627
add_to_lists(ce24, graph_objects, counter_examples, all_graphs)
3628
3629
# CE to independence_number(x) <= floor(e^(maximum(max_even_minus_even_horizontal(x), fiedler(x))))
3630
ce25 = Graph('OX??ZHEDxLvId_rgaC@SA')
3631
ce25.name(new = "ce25")
3632
add_to_lists(ce25, graph_objects, counter_examples, all_graphs)
3633
3634
# CE to independence_number(x) <= maximum(card_periphery(x), radius(x)*welsh_powell(x))
3635
ce26 = Graph("NF?_?o@?Oa?BC_?OOaO")
3636
ce26.name(new = "ce26")
3637
add_to_lists(ce26, graph_objects, counter_examples, all_graphs)
3638
3639
# CE to independence_number(x) <= floor(average_distance(x)) + maximum(max_even_minus_even_horizontal(x), brinkmann_steffen(x))
3640
ce27 = Graph("K_GBXS`ysCE_")
3641
ce27.name(new = "ce27")
3642
add_to_lists(ce27, graph_objects, counter_examples, all_graphs)
3643
3644
# CE to independence_number(x) <= minimum(annihilation_number(x), 2*e^sum_temperatures(x))
3645
ce28 = Graph("g??O?C_?`?@?O??A?A????????C?????G?????????A@aA??_???G??GA?@????????_???GHC???CG?_???@??_??OB?C?_??????_???G???C?O?????O??A??????G??")
3646
ce28.name(new = "ce28")
3647
add_to_lists(ce28, graph_objects, counter_examples, all_graphs)
3648
3649
# CE to independence_number(x) <= maximum(2*welsh_powell(x), maximum(max_even_minus_even_horizontal(x), laplacian_energy(x)))
3650
ce29 = Graph("P@g??BSCcIA???COcSO@@O@c")
3651
ce29.name(new = "ce29")
3652
add_to_lists(ce29, graph_objects, counter_examples, all_graphs)
3653
3654
# CE to independence_number(x) <= maximum(order_automorphism_group(x), 2*cvetkovic(x) - matching_number(x))
3655
ce30 = Graph("G~q|{W")
3656
ce30.name(new = "ce30")
3657
add_to_lists(ce30, graph_objects, counter_examples, all_graphs)
3658
3659
# CE to independence_number(x) <= max_even_minus_even_horizontal(x) + min_degree(x) + welsh_powell(x)
3660
ce31 = Graph("VP??oq_?PDOGhAwS??bSS_nOo?OHBqPi?I@AGP?POAi?")
3661
ce31.name(new = "ce31")
3662
add_to_lists(ce31, graph_objects, counter_examples, all_graphs)
3663
3664
# CE to independence_number(x) >= order(x)/szekeres_wilf(x)
3665
ce32 = Graph('H?`@Cbg')
3666
ce32.name(new = "ce32")
3667
add_to_lists(ce32, graph_objects, counter_examples, all_graphs)
3668
3669
# CE to independence_number(x) <= max_even_minus_even_horizontal(x) + minimum(card_positive_eigenvalues(x), card_center(x) + 1)
3670
ce33 = Graph("O_aHgP_kVSGOCXAiODcA_")
3671
ce33.name(new = "ce33")
3672
add_to_lists(ce33, graph_objects, counter_examples, all_graphs)
3673
3674
# CE to independence_number(x) <= card_center(x) + maximum(diameter(x), card_periphery(x))
3675
ce34 = Graph('H?PA_F_')
3676
ce34.name(new = "ce34")
3677
add_to_lists(ce34, graph_objects, counter_examples, all_graphs)
3678
3679
# CE to independence_number(x) <= card_center(x) + maximum(diameter(x), card_periphery(x))ce35 = Graph("")
3680
ce35 = Graph("HD`cgGO")
3681
ce35.name(new = "ce35")
3682
add_to_lists(ce35, graph_objects, counter_examples, all_graphs)
3683
3684
# CE to independence_number(x) >= max_degree(x) - order_automorphism_group(x)
3685
ce36 = Graph('ETzw')
3686
ce36.name(new = "ce36")
3687
add_to_lists(ce36, graph_objects, counter_examples, all_graphs)
3688
3689
# CE to independence_number(x) <= maximum(card_center(x), diameter(x)*max_degree(x))
3690
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?")
3691
ce37.name(new = "ce37")
3692
add_to_lists(ce37, graph_objects, counter_examples, all_graphs)
3693
3694
# CE to independence_number(x) <= abs(-card_center(x) + min_degree(x)) + max_even_minus_even_horizontal(x)
3695
ce38 = Graph('FVS_O')
3696
ce38.name(new = "ce38")
3697
add_to_lists(ce38, graph_objects, counter_examples, all_graphs)
3698
3699
# CE to independence_number(x) <= abs(-card_center(x) + max_degree(x)) + max_even_minus_even_horizontal(x)
3700
ce39 = Graph("FBAuo")
3701
ce39.name(new = "ce39")
3702
add_to_lists(ce39, graph_objects, counter_examples, all_graphs)
3703
3704
# CE to independence_number(x) <= floor(inverse_degree(x)) + order_automorphism_group(x) + 1
3705
ce40 = Graph('Htji~Ei')
3706
ce40.name(new = "ce40")
3707
add_to_lists(ce40, graph_objects, counter_examples, all_graphs)
3708
3709
# CE to independence_number(x) <= card_center(x) + maximum(residue(x), card_periphery(x))
3710
ce42 = Graph('GP[KGC')
3711
ce42.name(new = "ce42")
3712
add_to_lists(ce42, graph_objects, counter_examples, all_graphs)
3713
3714
# CE to independence_number(x) <= maximum(girth(x), (barrus_bound(x) - order_automorphism_group(x))^2)
3715
ce43 = Graph("Exi?")
3716
ce43.name(new = "ce43")
3717
add_to_lists(ce43, graph_objects, counter_examples, all_graphs)
3718
3719
# CE to independence_number(x) <= (brinkmann_steffen(x) - szekeres_wilf(x))^2 + max_even_minus_even_horizontal(x)
3720
ce44 = Graph('GGDSsg')
3721
ce44.name(new = "ce44")
3722
add_to_lists(ce44, graph_objects, counter_examples, all_graphs)
3723
3724
# CE to independence_number(x) <= maximum(max_even_minus_even_horizontal(x), radius(x)*szekeres_wilf(x))
3725
ce45 = Graph("FWKH?")
3726
ce45.name(new = "ce45")
3727
add_to_lists(ce45, graph_objects, counter_examples, all_graphs)
3728
3729
# CE to independence_number(x) <= maximum(card_periphery(x), radius(x)*szekeres_wilf(x))
3730
ce46 = Graph('F`I`?')
3731
ce46.name(new = "ce46")
3732
add_to_lists(ce46, graph_objects, counter_examples, all_graphs)
3733
3734
# CE to independence_number(x) <= maximum(card_periphery(x), diameter(x) + inverse_degree(x))
3735
ce47 = Graph("KVOzWAxewcaE")
3736
ce47.name(new = "ce47")
3737
add_to_lists(ce47, graph_objects, counter_examples, all_graphs)
3738
3739
# CE to independence_number(x) <= maximum(card_periphery(x), max_even_minus_even_horizontal(x) + min_degree(x))
3740
ce48 = Graph('Iq]ED@_s?')
3741
ce48.name(new = "ce48")
3742
add_to_lists(ce48, graph_objects, counter_examples, all_graphs)
3743
3744
# CE to independence_number(x) >= sqrt(card_positive_eigenvalues(x))
3745
ce49 = Graph("K^~lmrvv{~~Z")
3746
ce49.name(new = "ce49")
3747
add_to_lists(ce49, graph_objects, counter_examples, all_graphs)
3748
3749
# CE to independence_number(x) <= max_degree(x) + maximum(max_even_minus_even_horizontal(x), sigma_2(x))
3750
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??')
3751
ce50.name(new = "ce50")
3752
add_to_lists(ce50, graph_objects, counter_examples, all_graphs)
3753
3754
# CE to independence_number(x) >= matching_number(x) - order_automorphism_group(x) - 1
3755
ce51 = Graph("Ivq~j^~vw")
3756
ce51.name(new = "ce51")
3757
add_to_lists(ce51, graph_objects, counter_examples, all_graphs)
3758
3759
# CE to independence_number(x) >= order(x)/szekeres_wilf(x)
3760
ce52 = Graph('H?QaOiG')
3761
ce52.name(new = "ce52")
3762
add_to_lists(ce52, graph_objects, counter_examples, all_graphs)
3763
3764
# CE to independence_number(x) >= matching_number(x) - sigma_2(x) - 1
3765
ce53 = Graph("]?GEPCGg]S?`@??_EM@OTp?@E_gm?GW_og?pWO?_??GQ?A?^HIRwH?Y?__BC?G?[PD@Gs[O?GW")
3766
ce53.name(new = "ce53")
3767
add_to_lists(ce53, graph_objects, counter_examples, all_graphs)
3768
3769
# CE to independence_number(x) >= -average_distance(x) + ceil(lovasz_theta(x))
3770
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')
3771
ce54.name(new = "ce54")
3772
add_to_lists(ce54, graph_objects, counter_examples, all_graphs)
3773
3774
# CE to independence_number(x) >= -card_periphery(x) + matching_number(x)
3775
ce55 = Graph("I~~~~~~zw")
3776
ce55.name(new = "ce55")
3777
add_to_lists(ce55, graph_objects, counter_examples, all_graphs)
3778
3779
# CE to independence_number(x) >= lovasz_theta(x)/edge_con(x)
3780
ce56 = Graph('HsaGpOe')
3781
ce56.name(new = "ce56")
3782
add_to_lists(ce56, graph_objects, counter_examples, all_graphs)
3783
3784
# CE to independence_number(x) >= minimum(max_degree(x), floor(lovasz_theta(x)))
3785
ce57 = Graph("^?H{BDHqHosG??OkHOhE??B[CInU?@j_A?CoA^azGPLcb_@GEYYRPgG?K@gdPAg?d@_?_sGcED`@``O")
3786
ce57.name(new = "ce57")
3787
add_to_lists(ce57, graph_objects, counter_examples, all_graphs)
3788
3789
# CE to independence_number>= barrus_bound(x) - max(card_center(x), card_positive_eigenvalues(x))
3790
ce58 = Graph('Sj[{Eb~on~nls~NJWLVz~~^|{l]b\uFss')
3791
ce58.name(new = "ce58")
3792
add_to_lists(ce58, graph_objects, counter_examples, all_graphs)
3793
3794
# CE to independence_number(x) >= floor(tan(barrus_bound(x) - 1))
3795
ce59 = Graph("RxCWGCB?G?_B?@??_?N??F??B_??w?")
3796
ce59.name(new = "ce59")
3797
add_to_lists(ce59, graph_objects, counter_examples, all_graphs)
3798
3799
# CE to independence_number(x) >= -1/2*diameter(x) + lovasz_theta(x)
3800
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')
3801
ce60.name(new = "ce60")
3802
add_to_lists(ce60, graph_objects, counter_examples, all_graphs)
3803
3804
# CE to independence_number(x) <= maximum(card_negative_eigenvalues(x), max_common_neighbors(x) + max_even_minus_even_horizontal(x))
3805
ce61 = Graph("KsaAA?OOC??C")
3806
ce61.name(new = "ce61")
3807
add_to_lists(ce61, graph_objects, counter_examples, all_graphs)
3808
3809
# CE to independence_number(x) >= minimum(floor(lovasz_theta(x)), tan(spanning_trees_count(x)))
3810
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?")
3811
ce62.name(new = "ce62")
3812
add_to_lists(ce62, graph_objects, counter_examples, all_graphs)
3813
3814
# CE to independence_number(x) >= diameter(x)/different_degrees(x)
3815
ce63 = Graph("KOGkYBOCOAi@")
3816
ce63.name(new = "ce63")
3817
add_to_lists(ce63, graph_objects, counter_examples, all_graphs)
3818
3819
# CE to independence_number(x) >= -max_common_neighbors(x) + min_degree(x)
3820
ce64 = Graph('`szvym|h~RMQLTNNiZzsgQynDR\p~~rTZXi~n`kVvKolVJfP}TVEN}Thj~tv^KJ}D~VqqsNy|NY|ybklZLnz~TfyG')
3821
ce64.name(new = "ce64")
3822
add_to_lists(ce64, graph_objects, counter_examples, all_graphs)
3823
3824
# CE to independence_number(x) >= -10^different_degrees(x) + matching_number(x)
3825
ce65 = Graph("W~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~")
3826
ce65.name(new = "ce65")
3827
add_to_lists(ce65, graph_objects, counter_examples, all_graphs)
3828
3829
# CE to independence_number(x) >= girth^max_degree+1
3830
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??")
3831
ce66.name(new = "ce66")
3832
add_to_lists(ce66, graph_objects, counter_examples, all_graphs)
3833
3834
# CE to independence_number(x) <= maximum(cycle_space_dimension(x), floor(lovasz_theta(x)))
3835
ce67 = Graph("G??EDw")
3836
ce67.name(new = "ce67")
3837
add_to_lists(ce67, graph_objects, counter_examples, all_graphs)
3838
3839
# CE to independence_number(x) >= minimum(card_positive_eigenvalues(x), 2*card_zero_eigenvalues(x))
3840
ce68 = Graph('HzzP|~]')
3841
ce68.name(new = "ce68")
3842
add_to_lists(ce68, graph_objects, counter_examples, all_graphs)
3843
3844
# CE to independence_number(x) <= maximum(max_degree(x), radius(x)^card_periphery(x))
3845
ce69 = Graph("F?BvO")
3846
ce69.name(new = "ce69")
3847
add_to_lists(ce69, graph_objects, counter_examples, all_graphs)
3848
3849
# CE to independence_number(x) >= floor(lovasz_theta(x))/vertex_con(x)
3850
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_??????')
3851
ce70.name(new = "ce70")
3852
add_to_lists(ce70, graph_objects, counter_examples, all_graphs)
3853
3854
# CE to independence_number(x) <= maximum(matching_number(x), critical_independence_number(x))
3855
ce71 = Graph('ECYW')
3856
ce71.name(new = "ce71")
3857
add_to_lists(ce71, graph_objects, counter_examples, all_graphs)
3858
3859
# CE to independence_number(x)>=-1/2*x.diameter() + x.lovasz_theta()
3860
ce72 = Graph('fdSYkICGVs_m_TPs`Fmj_|pGhC@@_[@xWawsgEDe_@g`TC{P@pqGoocqOw?HBDS[R?CdG\e@kMCcgqr?G`NHGXgYpVGCoJdOKBJQAsG|ICE_BeMQGOwKqSd\W?CRg')
3861
ce72.name(new = "ce72")
3862
add_to_lists(ce72, graph_objects, counter_examples, all_graphs)
3863
3864
# CE to independence_number(x) >= minimum(floor(lovasz_theta(x)), max_even_minus_even_horizontal(x) + 1)
3865
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_@??')
3866
ce73.name(new = "ce73")
3867
add_to_lists(ce73, graph_objects, counter_examples, all_graphs)
3868
3869
# CE to independence_number(x) >= minimum(diameter(x), lovasz_theta(x))
3870
ce74 = Graph("FCQb_")
3871
ce74.name(new = "ce74")
3872
add_to_lists(ce74, graph_objects, counter_examples, all_graphs)
3873
3874
# CE to independence_number(x) >= minimum(girth(x), floor(lovasz_theta(x)))
3875
ce75 = Graph('E?Bw')
3876
ce75.name(new = "ce75")
3877
add_to_lists(ce75, graph_objects, counter_examples, all_graphs)
3878
3879
# CE to independence_number(x) <= maximum(average_distance(x), max_even_minus_even_horizontal(x))*sum_temperatures(x)
3880
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?")
3881
ce76.name(new = "ce76")
3882
add_to_lists(ce76, graph_objects, counter_examples, all_graphs)
3883
3884
# CE to independence_number(x) <= maximum(matching_number(x), critical_independence_number(x))
3885
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??")
3886
ce77.name(new = "ce77")
3887
add_to_lists(ce77, graph_objects, counter_examples, all_graphs)
3888
3889
# CE to independence_number(x) <= maximum(max_degree(x), radius(x)^card_periphery(x))
3890
ce78 = Graph("G_aCp[")
3891
ce78.name(new = "ce78")
3892
add_to_lists(ce78, graph_objects, counter_examples, all_graphs)
3893
3894
# CE to independence_number(x) <= residue(x)^2
3895
ce79 = Graph('J?B|~fpwsw_')
3896
ce79.name(new = "ce79")
3897
add_to_lists(ce79, graph_objects, counter_examples, all_graphs)
3898
3899
# CE to independence_number(x) <= 10^(card_center(x)*log(10)/log(sigma_2(x)))
3900
ce80 = Graph('T?????????????????F~~~v}~|zn}ztn}zt^')
3901
ce80.name(new = "ce80")
3902
add_to_lists(ce80, graph_objects, counter_examples, all_graphs)
3903
3904
# CE to independence_number(x) <= diameter(x)^card_periphery(x)
3905
ce81 = Graph('P?????????^~v~V~rzyZ~du{')
3906
ce81.name(new = "ce81")
3907
add_to_lists(ce81, graph_objects, counter_examples, all_graphs)
3908
3909
# CE to independence_number(x) <= radius(x)*residue(x) + girth(x)
3910
ce82 = Graph('O????B~~^Zx^wnc~ENqxY')
3911
ce82.name(new = "ce82")
3912
add_to_lists(ce82, graph_objects, counter_examples, all_graphs)
3913
3914
"""
3915
CE to independence_number(x) <= minimum(lovasz_theta(x), residue(x)^2)
3916
and
3917
<= minimum(annihilation_number(x), residue(x)^2)
3918
and
3919
<= minimum(fractional_alpha(x), residue(x)^2)
3920
and
3921
<= minimum(cvetkovic(x), residue(x)^2)
3922
and
3923
<= minimum(residue(x)^2, floor(lovasz_theta(x)))
3924
and
3925
<= minimum(size(x), residue(x)^2)
3926
"""
3927
ce83 = Graph('LEYSrG|mrQ[ppi')
3928
ce83.name(new = "ce83")
3929
add_to_lists(ce83, graph_objects, counter_examples, all_graphs)
3930
3931
# CE to independence_number(x) <= maximum(laplacian_energy(x), brinkmann_steffen(x)^2)
3932
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?')
3933
ce84.name(new = "ce84")
3934
add_to_lists(ce84, graph_objects, counter_examples, all_graphs)
3935
3936
# CE to independence_number(x) <= girth(x)^ceil(laplacian_energy(x))
3937
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?')
3938
ce85.name(new = "ce85")
3939
add_to_lists(ce85, graph_objects, counter_examples, all_graphs)
3940
3941
# CE to independence_number(x) <= diameter(x)*residue(x) + different_degrees(x)
3942
ce86 = Graph('SK|KWYc|^BJKlaCnMH^ECUoSC[{LHxfMG')
3943
ce86.name(new = "ce86")
3944
add_to_lists(ce86, graph_objects, counter_examples, all_graphs)
3945
3946
# CE to independence_number(x) <= maximum(max_common_neighbors(x), girth(x)^laplacian_energy(x))
3947
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_')
3948
ce87.name(new = "ce87")
3949
add_to_lists(ce87, graph_objects, counter_examples, all_graphs)
3950
3951
# CE to independence_number(x) <= maximum(radius(x), max_degree(x))^2
3952
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')
3953
ce88.name(new = "ce88")
3954
add_to_lists(ce88, graph_objects, counter_examples, all_graphs)
3955
3956
# CE to independence_number(x) <= max_degree(x) + maximum(max_even_minus_even_horizontal(x), geometric_length_of_degree_sequence(x))
3957
ce89 = Graph("_qH?S@??`??GG??O?_?C?_??@??@??G??C??_??C????O??G???@????O???A???@????C???C?????G????")
3958
ce89.name(new = "ce89")
3959
add_to_lists(ce89, graph_objects, counter_examples, all_graphs)
3960
3961
# CE to independence_number(x) >= floor(arccosh(lovasz_theta(x)))^2
3962
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")
3963
ce90.name(new = "ce90")
3964
add_to_lists(ce90, graph_objects, counter_examples, all_graphs)
3965
3966
# CE to independence_number(x) <= maximum(2*welsh_powell(x), max_even_minus_even_horizontal(x)^2)
3967
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???_??")
3968
ce91.name(new = "ce91")
3969
add_to_lists(ce91, graph_objects, counter_examples, all_graphs)
3970
3971
# CE to independence_number(x) >= -max_common_neighbors(x) + min_degree(x) - 1
3972
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?")
3973
ce92.name(new = "ce92")
3974
add_to_lists(ce92, graph_objects, counter_examples, all_graphs)
3975
3976
# CE to independence_number(x) >= diameter for regular graphs
3977
# Ryan Pepper, June 2017
3978
ce93 = Graph('_t???CA???_B?E?@??WAB?G??_GA?W?????@???W??B???A???BA??@__???G??@A???LA???AW???@_???G')
3979
ce93.name(new = "ce93")
3980
add_to_lists(ce93, graph_objects, counter_examples, all_graphs)
3981
3982
# CE to independence_number(x) >= diameter for regular graphs
3983
# Ryan Pepper, June 2017
3984
ce94 = Graph('Yv?GW?@?WB?A?A?@_?oGA?_KG????G??K??A???W??@???AO??BG??B?')
3985
ce94.name(new = "ce94")
3986
add_to_lists(ce94, graph_objects, counter_examples, all_graphs)
3987
3988
# CE to independence_number(x) >= -average_distance(x) + ceil(lovasz_theta(x))
3989
ce95 = Graph('epCpih@K}gSGIZfc?Rkf{EWtVKJTmJtWYl_IoDOKOikwDSKtbH\\fi_g`affO\\|Agq`WcLoakSLNPaWZ@PQhCh?ylTR\\tIR?WfoJNYJ@B{GiOWMUZX_puFP?')
3990
ce95.name(new = "ce95")
3991
add_to_lists(ce95, graph_objects, counter_examples, all_graphs)
3992
3993
# CE to independence_number(x) >= ceil(1/2*cvetkovic(x))
3994
# CE to independence_number(x) >= 1/2*cvetkovic(x)
3995
ce96 = Graph('Gvz~r{')
3996
ce96.name(new = "ce96")
3997
add_to_lists(ce96, graph_objects, counter_examples, all_graphs)
3998
3999
# CE to independence_number(x) >= -max_common_neighbors(x) + min_degree(x)
4000
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_')
4001
ce97.name(new = "ce97")
4002
add_to_lists(ce97, graph_objects, counter_examples, all_graphs)
4003
4004
# CE to independence_number(x) >= lovasz_theta(x)/radius(x)
4005
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')
4006
ce100.name(new = "ce100")
4007
add_to_lists(ce100, graph_objects, counter_examples, all_graphs)
4008
4009
# CE to independence_number(x) >= matching_number(x) - order_automorphism_group(x) - 1
4010
ce101 = Graph('I~~Lt~\Nw')
4011
ce101.name(new = "ce101")
4012
add_to_lists(ce101, graph_objects, counter_examples, all_graphs)
4013
4014
# CE to independence_number(x) >= card_positive_eigenvalues(x) - lovasz_theta(x)
4015
ce102 = Graph('N^nN~~}Z~|}~~\~]zzw')
4016
ce102.name(new = "ce102")
4017
add_to_lists(ce102, graph_objects, counter_examples, all_graphs)
4018
4019
# CE to independence_number(x) >= card_negative_eigenvalues(x) - sigma_2(x)
4020
ce103 = Graph('IOilnemjG')
4021
ce103.name(new = "ce103")
4022
add_to_lists(ce103, graph_objects, counter_examples, all_graphs)
4023
4024
# CE to independence_number(x) >= minimum(min_degree(x), floor(lovasz_theta(x)))
4025
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?')
4026
ce105.name(new = "ce105")
4027
add_to_lists(ce105, graph_objects, counter_examples, all_graphs)
4028
4029
# CE to independence_number(x) <= minimum(floor(lovasz_theta(x)), 10^laplacian_energy(x))
4030
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")
4031
ce106.name(new = "ce106")
4032
add_to_lists(ce106, graph_objects, counter_examples, all_graphs)
4033
4034
# CE to independence_number(x) <= max_even_minus_even_horizontal(x) + sinh(max_degree(x) - 1)
4035
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????_?@????")
4036
ce107.name(new = "ce107")
4037
add_to_lists(ce107, graph_objects, counter_examples, all_graphs)
4038
4039
# CE to independence_number(x) <= 10^cosh(log(residue(x)) - 1)
4040
ce108 = Graph("U??????g}yRozOzw\wBn?zoBv_FN?Bn?B|??vO??")
4041
ce108.name(new = "ce108")
4042
add_to_lists(ce108, graph_objects, counter_examples, all_graphs)
4043
4044
# CE to independence_number(x) >= floor(caro_wei(x))^min_common_neighbors(x)
4045
ce109 = Graph("Z???????????????????????????????B~~~v~~z~~|~nvnnjz~~xfntuV~w")
4046
ce109.name(new = "ce109")
4047
add_to_lists(ce109, graph_objects, counter_examples, all_graphs)
4048
4049
# CE to independence_number(x) >= min(card_positive_eigenvalues(x), 2*card_zero_eigenvalues(x))
4050
ce110 = Graph("GUxvuw")
4051
ce110.name(new = "ce110")
4052
add_to_lists(ce110, graph_objects, counter_examples, all_graphs)
4053
4054
# CE to independence_number(x) >= minimum(lovasz_theta(x), max_degree(x)/card_center(x))
4055
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")
4056
ce112.name(new = "ce112")
4057
add_to_lists(ce112, graph_objects, counter_examples, all_graphs)
4058
4059
# CE to independence_number(x) <= maximum(e^average_distance(x), max_degree(x)^2)
4060
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???_?????")
4061
ce114.name(new = "ce114")
4062
add_to_lists(ce114, graph_objects, counter_examples, all_graphs)
4063
4064
# CE to independence_number(x) <= 2*wilf(x)^2 + radius(x)
4065
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???_???????")
4066
ce115.name(new = "ce115")
4067
add_to_lists(ce115, graph_objects, counter_examples, all_graphs)
4068
4069
# CE to independence_number(x) <= card_periphery(x) + floor(e^laplacian_energy(x))
4070
ce116 = Graph("J?Bzvrw}Fo?")
4071
ce116.name(new = "ce116")
4072
add_to_lists(ce116, graph_objects, counter_examples, all_graphs)
4073
4074
# CE to independence_number(x) <= residue(x)^2/density(x)
4075
ce117 = Graph("N???F~}~f{^_z_~_^o?")
4076
ce117.name(new = "ce117")
4077
add_to_lists(ce117, graph_objects, counter_examples, all_graphs)
4078
4079
# CE to independence_number(x) <= maximum(caro_wei(x), e^different_degrees(x))^2
4080
ce118 = Graph("O????B~~v}^w~o~o^wF}?")
4081
ce118.name(new = "ce118")
4082
add_to_lists(ce118, graph_objects, counter_examples, all_graphs)
4083
4084
# CE to independence_number(x) <= 10^sinh(average_vertex_temperature(x) + card_center(x))
4085
ce119 = Graph("\?@CoO_CDOC?G?_?O?CO?_?BO?C???O?Bw??k??D_??O???_???C???A???@????g????")
4086
ce119.name(new = "ce119")
4087
add_to_lists(ce119, graph_objects, counter_examples, all_graphs)
4088
4089
# CE to independence_number(x) <= maximum(girth(x), min_degree(x))^laplacian_energy(x)
4090
ce120 = Graph("^????????????????????X~~Umf|mezrBcjoezwF^{_Un}?|w{?kQ[?@Z}?FNl??Etw??Oz??F~Z_??")
4091
ce120.name(new = "ce120")
4092
add_to_lists(ce120, graph_objects, counter_examples, all_graphs)
4093
4094
# CE to independence_number(x) <= (2*girth(x))^card_periphery(x)
4095
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???????????_??????????@???????????@")
4096
ce121.name(new = "ce121")
4097
add_to_lists(ce121, graph_objects, counter_examples, all_graphs)
4098
4099
# CE to independence_number(x) <= ceil(laplacian_energy(x)*max_degree(x))
4100
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??????")
4101
ce122.name(new = "ce122")
4102
add_to_lists(ce122, graph_objects, counter_examples, all_graphs)
4103
4104
# CE to independence_number(x) <= minimum(cvetkovic(x), 10^laplacian_energy(x))
4105
ce123 = Graph("`??????????????????????????????????????????^^}nx~~[~~x}^k~~F~v^o^~vm@^~~wF~v}_F~~^_B~~~W?")
4106
ce123.name(new = "ce123")
4107
add_to_lists(ce123, graph_objects, counter_examples, all_graphs)
4108
4109
# CE to independence_number(x) <= minimum(fractional_alpha(x), 10^laplacian_energy(x))
4110
ce124 = Graph("Z????????????????????~~~~~v~vf|~b~~o~~sF~~_^~}?~|k?z~w?^~}??")
4111
ce124.name(new = "ce124")
4112
add_to_lists(ce124, graph_objects, counter_examples, all_graphs)
4113
4114
# CE to independence_number(x) <= ceil(e^caro_wei(x))
4115
ce125 = Graph("L??F~z{~FwN_~?")
4116
ce125.name(new = "ce125")
4117
add_to_lists(ce125, graph_objects, counter_examples, all_graphs)
4118
4119
# CE to independence_number(x) <= ceil(e^inverse_degree(x))
4120
ce126 = Graph("Ov~~~}Lvs]^~~~~t~~}yJ")
4121
ce126.name(new = "ce126")
4122
add_to_lists(ce126, graph_objects, counter_examples, all_graphs)
4123
4124
# CE to independence_number(x) <= minimum(annihilation_number(x), 10^laplacian_energy(x))
4125
ce127 = Graph("[??????~~~^{~w~w^{F~?~WB~_F~?F~?B~_?~w?F~??^s??~w??~w??^{??F~???")
4126
ce127.name(new = "ce127")
4127
add_to_lists(ce127, graph_objects, counter_examples, all_graphs)
4128
4129
# CE to independence_number(x) <= (diameter(x) + 1)*residue(x)
4130
ce128 = Graph("M????B~~v}^w~o~o?")
4131
ce128.name(new = "ce128")
4132
add_to_lists(ce128, graph_objects, counter_examples, all_graphs)
4133
4134
# CE to independence_number(x) <= floor(10^sqrt(average_distance(x)))
4135
ce129 = Graph("X????????????????????????????F~~~~z~~~Z~~{n}|u~utn~")
4136
ce129.name(new = "ce129")
4137
add_to_lists(ce129, graph_objects, counter_examples, all_graphs)
4138
4139
# CE to independence_number(x) <= minimum(lovasz_theta(x), 10^laplacian_energy(x))
4140
ce130 = Graph("]?????????????????F|~nzz~~f~~F|~B~~_~~WF~~?^~{?nng?~~w?^^{?F~n??~~w?@~~_??")
4141
ce130.name(new = "ce130")
4142
add_to_lists(ce130, graph_objects, counter_examples, all_graphs)
4143
4144
# CE to independence_number(x) <= (residue(x) + 1)*girth(x)
4145
ce131 = Graph("W?????????^~~}~}^~F~o~}B~wF}oF~oB~w?~}?F~o?^~??")
4146
ce131.name(new = "ce131")
4147
add_to_lists(ce131, graph_objects, counter_examples, all_graphs)
4148
4149
# CE to independence_number(x) <= minimum(size(x), 10^laplacian_energy(x))
4150
ce132 = Graph("b????????????????????~~|~~|}~x~n~vp~{Nz{^e~svN}Xc~zFKZ~uyX[z[J\\ti@M~mT@]}X{SzQZQQ^posGNlZg?R}lp`LX{L?")
4151
ce132.name(new = "ce132")
4152
add_to_lists(ce132, graph_objects, counter_examples, all_graphs)
4153
4154
# CE to independence_number(x) >= -brinkmann_steffen(x) - girth(x) + matching_number(x)
4155
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")
4156
ce133.name(new = "ce133")
4157
add_to_lists(ce133, graph_objects, counter_examples, all_graphs)
4158
4159
# CE to independence_number(x) >= -max_common_neighbors(x) + min_degree(x) - 1
4160
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')
4161
ce134.name(new = "ce134")
4162
add_to_lists(ce134, graph_objects, counter_examples, all_graphs)
4163
4164
# CE to independence_number(x) >= floor(tan(brooks(x)))
4165
ce135 = Graph('Nzx~VT}yzxNd^J^Jn~w')
4166
ce135.name(new = "ce135")
4167
add_to_lists(ce135, graph_objects, counter_examples, all_graphs)
4168
4169
# CE to independence_number(x) >= floor(lovasz_theta(x))*sin(gutman_energy(x))
4170
paley_17 = graphs.PaleyGraph(17)
4171
paley_17.name(new = "paley_17")
4172
add_to_lists(paley_17, graph_objects, counter_examples, all_graphs)
4173
4174
# CE to independence_number(x) >= ceil(lovasz_theta(x)) - diameter(x)
4175
# CE to independence_number(x) >= ceil(lovasz_theta(x)) - radius(x)
4176
paley_37 = graphs.PaleyGraph(37)
4177
paley_37.name(new = "paley_37")
4178
add_to_lists(paley_37, graph_objects, counter_examples, all_graphs)
4179
4180
# CE to independence_number(x) >= minimum(floor(lovasz_theta(x)), tan(randic(x)))
4181
paley_53e = graphs.PaleyGraph(53)
4182
paley_53e.add_edge(52,53)
4183
paley_53e.name(new = "paley_53e")
4184
add_to_lists(paley_53e, graph_objects, counter_examples, all_graphs)
4185
4186
paley_101 = graphs.PaleyGraph(101)
4187
paley_101.name(new = "paley_101")
4188
add_to_lists(paley_101, graph_objects, all_graphs)
4189
4190
# CE to independence_number(x) >= cos(alon_spencer(x))*floor(lovasz_theta(x))
4191
paley_149 = graphs.PaleyGraph(149)
4192
paley_149.name(new = "paley_149")
4193
add_to_lists(paley_149, graph_objects, counter_examples, all_graphs)
4194
4195
#a K5 with a pendant
4196
# CE to dirac => regular or planar
4197
k5pendant = Graph('E~}?')
4198
k5pendant.name(new="k5pendant")
4199
add_to_lists(k5pendant, graph_objects, counter_examples, all_graphs)
4200
4201
#alon_seymour graph:
4202
# CE to the rank-coloring conjecture,
4203
# 56-regular, vertex_trans, alpha=2, omega=22, chi=chi'=edge_connect=56
4204
alon_seymour=Graph([[0..63], lambda x,y : operator.xor(x,y) not in (0,1,2,4,8,16,32,63)])
4205
alon_seymour.name(new="alon_seymour")
4206
add_to_lists(alon_seymour, problem_graphs, counter_examples, all_graphs)
4207
4208
edge_critical_5=graphs.CycleGraph(5)
4209
edge_critical_5.add_edge(0,3)
4210
edge_critical_5.add_edge(1,4)
4211
edge_critical_5.name(new="edge_critical_5")
4212
add_to_lists(edge_critical_5, graph_objects, chromatic_index_critical, all_graphs)
4213
4214
# CE to independence_number(x) >= min(e - n + 1, diameter(x))
4215
heather = graphs.CompleteGraph(4)
4216
heather.add_vertex()
4217
heather.add_vertex()
4218
heather.add_edge(0,4)
4219
heather.add_edge(5,4)
4220
heather.name(new="heather")
4221
add_to_lists(heather, graph_objects, counter_examples, all_graphs)
4222
4223
#residue = alpha = 3
4224
# CE to residue = alpha => is_ore
4225
ryan3=graphs.CycleGraph(15)
4226
for i in range(15):
4227
for j in [1,2,3]:
4228
ryan3.add_edge(i,(i+j)%15)
4229
ryan3.add_edge(i,(i-j)%15)
4230
ryan3.name(new="ryan3")
4231
add_to_lists(ryan3, graph_objects, counter_examples, all_graphs)
4232
4233
#sylvester graph: 3-reg, 3 bridges, no perfect matching (why Petersen theorem requires no more than 2 bridges)
4234
sylvester = Graph('Olw?GCD@o??@?@?A_@o`A')
4235
sylvester.name(new="sylvester")
4236
add_to_lists(sylvester, graph_objects, all_graphs)
4237
4238
fork = graphs.PathGraph(4)
4239
fork.add_vertex()
4240
fork.add_edge(1,4)
4241
fork.name(new="fork")
4242
add_to_lists(fork, graph_objects, all_graphs)
4243
4244
# one of the 2 order 11 chromatic edge-critical graphs discovered by brinkmann and steffen
4245
edge_critical_11_1 = graphs.CycleGraph(11)
4246
edge_critical_11_1.add_edge(0,2)
4247
edge_critical_11_1.add_edge(1,6)
4248
edge_critical_11_1.add_edge(3,8)
4249
edge_critical_11_1.add_edge(5,9)
4250
edge_critical_11_1.name(new="edge_critical_11_1")
4251
add_to_lists(edge_critical_11_1, graph_objects, chromatic_index_critical, all_graphs)
4252
4253
#one of the 2 order 11 chromatic edge-critical graphs discovered by brinkmann and steffen
4254
edge_critical_11_2 = graphs.CycleGraph(11)
4255
edge_critical_11_2.add_edge(0,2)
4256
edge_critical_11_2.add_edge(3,7)
4257
edge_critical_11_2.add_edge(6,10)
4258
edge_critical_11_2.add_edge(4,9)
4259
edge_critical_11_2.name(new="edge_critical_11_2")
4260
add_to_lists(edge_critical_11_2, graph_objects, chromatic_index_critical, all_graphs)
4261
4262
# chromatic_index_critical but not overfull
4263
pete_minus=graphs.PetersenGraph()
4264
pete_minus.delete_vertex(9)
4265
pete_minus.name(new="pete_minus")
4266
add_to_lists(pete_minus, graph_objects, chromatic_index_critical, all_graphs)
4267
4268
"""
4269
The Haemers graph was considered by Haemers who showed that alpha(G)=theta(G)<vartheta(G).
4270
The graph is a 108-regular graph on 220 vertices. The vertices correspond to the 3-element
4271
subsets of {1,...,12} and two such vertices are adjacent whenever the subsets
4272
intersect in exactly one element.
4273
4274
sage: haemers
4275
haemers: Graph on 220 vertices
4276
sage: haemers.is_regular()
4277
True
4278
sage: max(haemers.degree())
4279
108
4280
"""
4281
haemers = Graph([Subsets(12,3), lambda s1,s2: len(s1.intersection(s2))==1])
4282
haemers.relabel()
4283
haemers.name(new="haemers")
4284
add_to_lists(haemers, problem_graphs, all_graphs)
4285
4286
"""
4287
The Pepper residue graph was described by Ryan Pepper in personal communication.
4288
It is a graph which demonstrates that the residue is not monotone. The graph is
4289
constructed by taking the complete graph on 3 vertices and attaching a pendant
4290
vertex to each of its vertices, then taking two copies of this graph, adding a
4291
vertex and connecting it to all the pendant vertices. This vertex has degree
4292
sequence [6, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2] which gives residue equal to 4.
4293
By removing the central vertex with degree 6, you get a graph with degree
4294
sequence [3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1] which has residue equal to 5.
4295
4296
sage: pepper_residue_graph
4297
pepper_residue_graph: Graph on 13 vertices
4298
sage: sorted(pepper_residue_graph.degree(), reverse=True)
4299
[6, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2]
4300
sage: residue(pepper_residue_graph)
4301
4
4302
sage: residue(pepper_residue_graph.subgraph(vertex_property=lambda v:pepper_residue_graph.degree(v)<6))
4303
5
4304
"""
4305
pepper_residue_graph = graphs.CompleteGraph(3)
4306
pepper_residue_graph.add_edges([(i,i+3) for i in range(3)])
4307
pepper_residue_graph = pepper_residue_graph.disjoint_union(pepper_residue_graph)
4308
pepper_residue_graph.add_edges([(0,v) for v in pepper_residue_graph.vertices() if pepper_residue_graph.degree(v)==1])
4309
pepper_residue_graph.relabel()
4310
pepper_residue_graph.name(new="pepper_residue_graph")
4311
add_to_lists(pepper_residue_graph, graph_objects, all_graphs)
4312
4313
"""
4314
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
4315
less than the independence number of any realization of the degree sequence. The degree sequence is [4^8,2].
4316
The realization is the one given by reversing the Havel-Hakimi process.
4317
4318
sage: barrus_graph
4319
barrus_graph: Graph on 9 vertices
4320
sage: residue(barrus_graph)
4321
2
4322
sage: independence_number(barrus_graph)
4323
3
4324
"""
4325
barrus_graph = Graph('HxNEG{W')
4326
barrus_graph.name(new = "barrus_graph")
4327
add_to_lists(barrus_graph, graph_objects, all_graphs)
4328
4329
# CE to (is_split)->((is_eulerian)->(is_regular))
4330
# split graph from k4 and e2 that is eulerian but not regular
4331
# isomorphic to c6 with a k4 subgraph
4332
# eulerain, diameter = 3, radius=2, hamiltonian
4333
4334
k4e2split = graphs.CompleteGraph(4)
4335
k4e2split.add_vertices([4,5])
4336
k4e2split.add_edge(4,0)
4337
k4e2split.add_edge(4,1)
4338
k4e2split.add_edge(5,2)
4339
k4e2split.add_edge(5,3)
4340
k4e2split.name(new = "k4e2split")
4341
add_to_lists(k4e2split, graph_objects, counter_examples, all_graphs)
4342
4343
# CE to (has_residue_equals_alpha)->((is_eulerian)->(alpha_leq_order_over_two))
4344
triangle_star = Graph("H}qdB@_")
4345
triangle_star.name(new = "triangle_star")
4346
add_to_lists(triangle_star, graph_objects, counter_examples, all_graphs)
4347
4348
#flower with n petals
4349
def flower(n):
4350
g = graphs.StarGraph(2*n)
4351
for x in range(n):
4352
v = 2*x+1
4353
g.add_edge(v,v+1)
4354
return g
4355
4356
flower_with_3_petals = flower(3)
4357
flower_with_3_petals.name(new = "flower_with_3_petals")
4358
add_to_lists(flower_with_3_petals, graph_objects, all_graphs)
4359
4360
flower_with_4_petals = flower(4)
4361
flower_with_4_petals.name(new = "flower_with_4_petals")
4362
add_to_lists(flower_with_4_petals, graph_objects, all_graphs)
4363
4364
# Gallai Tree graph
4365
gallai_tree = Graph("`hCKGC@?G@?K?@?@_?w?@??C??G??G??c??o???G??@_??F???N????_???G???B????C????W????G????G????C")
4366
gallai_tree.name(new = "gallai_tree")
4367
add_to_lists(gallai_tree, graph_objects, all_graphs)
4368
4369
# Trigonal Antiprism w/ capped top face
4370
trig_antiprism_capped = Graph("Iw?EthkF?")
4371
trig_antiprism_capped.name(new = "trig_antiprism_capped")
4372
add_to_lists(trig_antiprism_capped, graph_objects, all_graphs)
4373
4374
"""
4375
From Willis's thesis, page 4
4376
Alpha = Fractional Alpha = 4
4377
4378
sage: independence_number(willis_page4)
4379
4
4380
sage: fractional_alpha(willis_page4)
4381
4
4382
"""
4383
willis_page4 = Graph("GlCKIS")
4384
willis_page4.name(new = "willis_page4")
4385
add_to_lists(willis_page4, graph_objects, all_graphs)
4386
4387
"""
4388
From Willis's thesis, page 7
4389
4390
sage: willis_page7.radius()
4391
2
4392
sage: average_distance(willis_page7)
4393
1.467
4394
"""
4395
willis_page7 = Graph("ELRW")
4396
willis_page7.name(new = "willis_page7")
4397
add_to_lists(willis_page7, graph_objects, all_graphs)
4398
4399
"""
4400
From Willis's thesis, page 13, Fig. 2.7
4401
4402
sage: independence_number(willis_page13_fig27)
4403
4
4404
sage: willis_page13_fig27.order()
4405
7
4406
sage: willis_page13_fig27.size()
4407
15
4408
"""
4409
willis_page13_fig27 = Graph("Fs\zw")
4410
willis_page13_fig27.name(new = "willis_page13_fig27")
4411
add_to_lists(willis_page13_fig27, graph_objects, all_graphs)
4412
4413
"""
4414
From Willis's thesis, page 10, Figure 2.2
4415
Graph for which the Cvetkovic bound is the best upper bound present in the thesis
4416
4417
sage: independence_number(willis_page10_fig23)
4418
4
4419
sage: willis_page10_fig23.order()
4420
10
4421
sage: willis_page10_fig23.size()
4422
15
4423
sage: max_degree(willis_page10_fig23)
4424
3
4425
sage: min_degree(willis_page10_fig23)
4426
3
4427
"""
4428
willis_page10_fig23 = Graph("G|eKHw")
4429
willis_page10_fig23.name(new = "willis_page10_fig23")
4430
add_to_lists(willis_page10_fig23, graph_objects, all_graphs)
4431
4432
"""
4433
From Willis's thesis, page 10, Figure 2.4
4434
Graph for which the Cvetkovic bound is the best upper bound present in the thesis
4435
4436
sage: independence_number(willis_page10_fig24)
4437
9
4438
sage: willis_page10_fig24.order()
4439
24
4440
sage: willis_page10_fig24.size()
4441
36
4442
sage: max_degree(willis_page10_fig24)
4443
3
4444
sage: min_degree(willis_page10_fig24)
4445
3
4446
"""
4447
willis_page10_fig24 = Graph("WvOGWK@?G@_B???@_?O?F?????G??W?@K_?????G??@_?@B")
4448
willis_page10_fig24.name(new = "willis_page10_fig24")
4449
add_to_lists(willis_page10_fig24, graph_objects, all_graphs)
4450
4451
"""
4452
From Willis's thesis, page 13, Figure 2.6
4453
Graph for which the fractional independence bound is the best upper bound present in the thesis
4454
4455
sage: independence_number(willis_page13_fig26)
4456
3
4457
sage: willis_page13_fig26.order()
4458
7
4459
sage: willis_page13_fig26.size()
4460
12
4461
sage: max_degree(willis_page13_fig26)
4462
4
4463
sage: min_degree(willis_page13_fig26)
4464
3
4465
"""
4466
willis_page13_fig26 = Graph("FstpW")
4467
willis_page13_fig26.name(new = "willis_page13_fig26")
4468
add_to_lists(willis_page13_fig26, graph_objects, all_graphs)
4469
4470
"""
4471
From Willis's thesis, page 21, Figure 3.1
4472
Graph for which n/chi is the best lower bound present in the thesis
4473
4474
sage: independence_number(willis_page21)
4475
4
4476
sage: willis_page21.order()
4477
12
4478
sage: willis_page21.size()
4479
20
4480
sage: max_degree(willis_page21)
4481
4
4482
sage: chromatic_num(willis_page21)
4483
3
4484
"""
4485
willis_page21 = Graph("KoD?Xb?@HBBB")
4486
willis_page21.name(new = "willis_page21")
4487
add_to_lists(willis_page21, graph_objects, all_graphs)
4488
4489
"""
4490
From Willis's thesis, page 25, Figure 3.2
4491
Graph for which residue is the best lower bound present in the thesis
4492
4493
sage: independence_number(willis_page25_fig32)
4494
3
4495
sage: willis_page25_fig32.order()
4496
8
4497
sage: willis_page25_fig32.size()
4498
15
4499
sage: max_degree(willis_page25_fig32)
4500
6
4501
sage: chromatic_num(willis_page25_fig32)
4502
4
4503
"""
4504
willis_page25_fig32 = Graph("G@N@~w")
4505
willis_page25_fig32.name(new = "willis_page25_fig32")
4506
add_to_lists(willis_page25_fig32, graph_objects, all_graphs)
4507
4508
"""
4509
From Willis's thesis, page 25, Figure 3.3
4510
Graph for which residue is the best lower bound present in the thesis
4511
4512
sage: independence_number(willis_page25_fig33)
4513
4
4514
sage: willis_page25_fig33.order()
4515
14
4516
sage: willis_page25_fig33.size()
4517
28
4518
sage: max_degree(willis_page25_fig33)
4519
4
4520
sage: chromatic_num(willis_page25_fig33)
4521
4
4522
"""
4523
willis_page25_fig33 = Graph("Mts?GKE@QDCIQIKD?")
4524
willis_page25_fig33.name(new = "willis_page25_fig33")
4525
add_to_lists(willis_page25_fig33, graph_objects, all_graphs)
4526
4527
# The Lemke Graph
4528
lemke = Graph("G_?ztw")
4529
lemke.name(new = "Lemke")
4530
add_to_lists(lemke, graph_objects, all_graphs)
4531
4532
"""
4533
From Willis's thesis, page 29, Figure 3.6
4534
Graph for which the Harant Bound is the best lower bound present in the thesis
4535
4536
sage: independence_number(willis_page29)
4537
4
4538
sage: willis_page29.order()
4539
14
4540
sage: willis_page29.size()
4541
28
4542
sage: max_degree(willis_page29)
4543
4
4544
sage: chromatic_num(willis_page29)
4545
4
4546
"""
4547
willis_page29 = Graph("[HCGGC@?G?_@?@_?_?M?@o??_?G_?GO?CC?@?_?GA??_C?@?C?@?A??_?_?G?D?@")
4548
willis_page29.name(new = "willis_page29")
4549
add_to_lists(willis_page29, graph_objects, all_graphs)
4550
4551
"""
4552
From Willis's thesis, page 35, Figure 5.1
4553
A graph where none of the upper bounds in the thesis give the exact value for alpha
4554
4555
sage: independence_number(willis_page35_fig51)
4556
2
4557
sage: willis_page35_fig51.order()
4558
10
4559
"""
4560
willis_page35_fig51 = Graph("I~rH`cNBw")
4561
willis_page35_fig51.name(new = "willis_page35_fig51")
4562
add_to_lists(willis_page35_fig51, graph_objects, all_graphs)
4563
4564
"""
4565
From Willis's thesis, page 35, Figure 5.2
4566
A graph where none of the upper bounds in the thesis give the exact value for alpha
4567
4568
sage: independence_number(willis_page35_fig52)
4569
2
4570
sage: willis_page35_fig52.order()
4571
10
4572
"""
4573
willis_page35_fig52 = Graph("I~zLa[vFw")
4574
willis_page35_fig52.name(new = "willis_page35_fig52")
4575
add_to_lists(willis_page35_fig52, graph_objects, all_graphs)
4576
4577
"""
4578
From Willis's thesis, page 36, Figure 5.3
4579
A graph where none of the upper bounds in the thesis give the exact value for alpha
4580
4581
sage: independence_number(willis_page36_fig53)
4582
4
4583
sage: willis_page36_fig53.order()
4584
11
4585
"""
4586
willis_page36_fig53 = Graph("JscOXHbWqw?")
4587
willis_page36_fig53.name(new = "willis_page36_fig53")
4588
add_to_lists(willis_page36_fig53, graph_objects, all_graphs)
4589
4590
"""
4591
From Willis's thesis, page 36, Figure 5.4
4592
A graph where none of the upper bounds in the thesis give the exact value for alpha
4593
4594
sage: independence_number(willis_page36_fig54)
4595
2
4596
sage: willis_page36_fig54.order()
4597
9
4598
sage: willis_page36_fig54.size()
4599
24
4600
"""
4601
willis_page36_fig54 = Graph("H~`HW~~")
4602
willis_page36_fig54.name(new = "willis_page36_fig54")
4603
add_to_lists(willis_page36_fig54, graph_objects, all_graphs)
4604
4605
"""
4606
From Willis's thesis, page 36, Figure 5.5
4607
A graph where none of the lower bounds in the thesis give the exact value for alpha
4608
4609
sage: independence_number(willis_page36_fig55)
4610
3
4611
sage: willis_page36_fig54.order()
4612
7
4613
sage: willis_page36_fig54.size()
4614
13
4615
"""
4616
willis_page36_fig55 = Graph("F@^vo")
4617
willis_page36_fig55.name(new = "willis_page36_fig55")
4618
add_to_lists(willis_page36_fig55, graph_objects, all_graphs)
4619
4620
"""
4621
From Willis's thesis, page 37, Figure 5.6
4622
A graph where none of the lower bounds in the thesis give the exact value for alpha
4623
4624
sage: independence_number(willis_page37_fig56)
4625
3
4626
sage: willis_page37_fig56.order()
4627
7
4628
sage: willis_page37_fig56.size()
4629
15
4630
"""
4631
willis_page37_fig56 = Graph("Fimzw")
4632
willis_page37_fig56.name(new = "willis_page37_fig56")
4633
add_to_lists(willis_page37_fig56, graph_objects, all_graphs)
4634
4635
"""
4636
From Willis's thesis, page 37, Figure 5.8
4637
A graph where none of the lower bounds in the thesis give the exact value for alpha
4638
4639
sage: independence_number(willis_page37_fig58)
4640
3
4641
sage: willis_page37_fig58.order()
4642
9
4643
sage: willis_page37_fig58.size()
4644
16
4645
"""
4646
willis_page37_fig58 = Graph("H?iYbC~")
4647
willis_page37_fig58.name(new = "willis_page37_fig58")
4648
add_to_lists(willis_page37_fig58, graph_objects, all_graphs)
4649
4650
"""
4651
From Willis's thesis, page 38, Figure 5.9
4652
A graph where none of the lower bounds in the thesis give the exact value for alpha
4653
4654
sage: independence_number(willis_page38)
4655
3
4656
sage: willis_page38.order()
4657
7
4658
sage: willis_page38.size()
4659
13
4660
"""
4661
willis_page38 = Graph("FSpzw")
4662
willis_page38.name(new = "willis_page38")
4663
add_to_lists(willis_page38, graph_objects, all_graphs)
4664
4665
"""
4666
From Willis's thesis, page 39, Figure 5.10
4667
A graph where none of the upper or lower bounds in the thesis give the exact value for alpha
4668
4669
sage: independence_number(willis_page39_fig510)
4670
5
4671
sage: willis_page39_fig510.order()
4672
12
4673
sage: willis_page39_fig510.size()
4674
18
4675
"""
4676
willis_page39_fig510 = Graph("Kt?GOKEOGal?")
4677
willis_page39_fig510.name(new = "willis_page39_fig510")
4678
add_to_lists(willis_page39_fig510, graph_objects, all_graphs)
4679
4680
"""
4681
From Willis's thesis, page 40, Figure 5.12
4682
A graph where none of the upper or lower bounds in the thesis give the exact value for alpha
4683
4684
sage: independence_number(willis_page40_fig512)
4685
6
4686
sage: willis_page40_fig512.order()
4687
14
4688
sage: willis_page40_fig512.size()
4689
21
4690
"""
4691
willis_page40_fig512 = Graph("Ms???\?OGdAQJ?J??")
4692
willis_page40_fig512.name(new = "willis_page40_fig512")
4693
add_to_lists(willis_page40_fig512, graph_objects, all_graphs)
4694
4695
"""
4696
From Willis's thesis, page 40, Figure 5.13
4697
A graph where none of the upper or lower bounds in the thesis give the exact value for alpha
4698
"""
4699
willis_page40_fig513 = Graph("UhCGKE??Q?_D?K??__g?@G?C??OC???ACG?DC??_")
4700
willis_page40_fig513.name(new = "willis_page40_fig513")
4701
add_to_lists(willis_page40_fig513, graph_objects, all_graphs)
4702
4703
"""
4704
From Willis's thesis, page 41, Figure 5.14
4705
A graph where none of the upper or lower bounds in the thesis give the exact value for alpha
4706
4707
sage: independence_number(willis_page41_fig514)
4708
5
4709
sage: willis_page41_fig514.order()
4710
12
4711
sage: willis_page41_fig514.size()
4712
18
4713
"""
4714
willis_page41_fig514 = Graph("Kt?GGGBQGeL?")
4715
willis_page41_fig514.name(new = "willis_page41_fig514")
4716
add_to_lists(willis_page41_fig514, graph_objects, all_graphs)
4717
4718
"""
4719
From Willis's thesis, page 41, Figure 5.15
4720
A graph where none of the upper or lower bounds in the thesis give the exact value for alpha
4721
4722
sage: independence_number(willis_page41_fig515)
4723
4
4724
sage: willis_page41_fig515.order()
4725
11
4726
sage: willis_page41_fig515.size()
4727
22
4728
"""
4729
willis_page41_fig515 = Graph("JskIIDBLPh?")
4730
willis_page41_fig515.name(new = "willis_page41_fig515")
4731
add_to_lists(willis_page41_fig515, graph_objects, all_graphs)
4732
4733
"""
4734
From Elphick-Wocjan page 8
4735
"""
4736
elphick_wocjan_page8 = Graph("F?Azw")
4737
elphick_wocjan_page8.name(new = "Elphick-Wocjan p.8")
4738
add_to_lists(elphick_wocjan_page8, graph_objects, all_graphs)
4739
4740
"""
4741
From Elphick-Wocjan page 9
4742
"""
4743
elphick_wocjan_page9 = Graph("FqhXw")
4744
elphick_wocjan_page9.name(new = "Elphick-Wocjan p.9")
4745
add_to_lists(elphick_wocjan_page9, graph_objects, all_graphs)
4746
4747
"""
4748
An odd wheel with 8 vertices
4749
p.175
4750
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.
4751
4752
sage: odd_wheel_8.order()
4753
8
4754
sage: odd_wheel_8.size()
4755
14
4756
"""
4757
odd_wheel_8 = Graph("G|eKMC")
4758
odd_wheel_8.name(new = "odd_wheel_8")
4759
add_to_lists(odd_wheel_8, graph_objects, all_graphs)
4760
4761
"""
4762
A facet-inducing graph
4763
p.176
4764
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.
4765
4766
sage: facet_inducing.order()
4767
8
4768
sage: facet_inducing.size()
4769
11
4770
"""
4771
facet_inducing = Graph("G@hicc")
4772
facet_inducing.name(new = "facet_inducing")
4773
add_to_lists(facet_inducing, graph_objects, all_graphs)
4774
4775
"""
4776
Double Fork
4777
p.185
4778
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.
4779
4780
sage: double_fork.order()
4781
6
4782
sage: double_fork.size()
4783
5
4784
"""
4785
double_fork = Graph("E?dg")
4786
double_fork.name(new = "double_fork")
4787
add_to_lists(double_fork, graph_objects, all_graphs)
4788
4789
double_fork_sub = Graph("G??MPg")
4790
double_fork_sub.name(new = "double_fork_sub")
4791
add_to_lists(double_fork_sub, graph_objects, all_graphs)
4792
4793
"""
4794
Golomb Graph
4795
Appears in THE FRACTIONAL CHROMATIC NUMBER OF THE PLANE by Cranston and Rabern
4796
"""
4797
golomb = Graph("I?C]dPcww")
4798
golomb.name(new = "Golomb Graph")
4799
add_to_lists(golomb, graph_objects, all_graphs)
4800
4801
"""
4802
Fig. 1, p. 2 of
4803
Steinberg’s Conjecture is false by
4804
Vincent Cohen-Addad, Michael Hebdige, Daniel Král,
4805
Zhentao Li, Esteban Salgado
4806
"""
4807
steinberg_ce_g1 = Graph("N?CWGOOOH@OO_POdCHO")
4808
steinberg_ce_g1.name(new = "steinberg_ce_g1")
4809
add_to_lists(steinberg_ce_g1, graph_objects, counter_examples, all_graphs)
4810
4811
"""
4812
Fig. 2, p. 3 of
4813
Steinberg’s Conjecture is false by
4814
Vincent Cohen-Addad, Michael Hebdige, Daniel Král,
4815
Zhentao Li, Esteban Salgado
4816
"""
4817
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")
4818
steinberg_ce_g2.name(new = "steinberg_ce_g2")
4819
add_to_lists(steinberg_ce_g2, graph_objects, counter_examples, all_graphs)
4820
4821
"""
4822
Fig. 3, p. 4 of
4823
Steinberg’s Conjecture is false by
4824
Vincent Cohen-Addad, Michael Hebdige, Daniel Král,
4825
Zhentao Li, Esteban Salgado
4826
"""
4827
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????_??_")
4828
steinberg_ce_g3.name(new = "steinberg_ce_g3")
4829
add_to_lists(steinberg_ce_g3, graph_objects, counter_examples, all_graphs)
4830
4831
"""
4832
4-pan from p. 1691 of
4833
Graphs with the Strong Havel–Hakimi Property by Michael D. Barrus and Grant Molnar
4834
"""
4835
four_pan = Graph("DBw")
4836
four_pan.name(new = "4-pan")
4837
add_to_lists(four_pan, graph_objects, all_graphs)
4838
4839
"""
4840
kite from p. 1691 of
4841
Graphs with the Strong Havel–Hakimi Property by Michael D. Barrus and Grant Molnar
4842
NOTE: Called kite in the paper, but will be called kite_with_tail here because we already have a kite
4843
"""
4844
kite_with_tail = Graph("DJk")
4845
kite_with_tail.name(new = "kite with tail")
4846
add_to_lists(kite_with_tail, graph_objects, all_graphs)
4847
4848
"""
4849
Chartrand Fig 1.1
4850
4851
sage: chartrand_11.order()
4852
8
4853
sage: chartrand_11.size()
4854
15
4855
"""
4856
chartrand_11 = Graph("G`RHx{")
4857
chartrand_11.name(new = "chartrand fig 1.1")
4858
add_to_lists(chartrand_11, graph_objects, all_graphs)
4859
4860
"""
4861
Chartrand Fig 1.2
4862
4863
sage: chartrand_12.order()
4864
8
4865
sage: chartrand_12.size()
4866
9
4867
"""
4868
chartrand_12 = Graph("G??|Qo")
4869
chartrand_12.name(new = "chartrand fig 1.2")
4870
add_to_lists(chartrand_12, graph_objects, all_graphs)
4871
4872
"""
4873
Chartrand Fig 1.3
4874
4875
sage: chartrand_13.order()
4876
8
4877
sage: chartrand_13.size()
4878
10
4879
"""
4880
chartrand_13 = Graph("G`o_g[")
4881
chartrand_13.name(new = "chartrand fig 1.3")
4882
add_to_lists(chartrand_13, graph_objects, all_graphs)
4883
4884
"""
4885
Chartrand Fig 1.8 - G
4886
4887
sage: chartrand_18_g.order()
4888
7
4889
sage: chartrand_18_g.size()
4890
8
4891
"""
4892
chartrand_18_g = Graph("Fo@Xo")
4893
chartrand_18_g.name(new = "chartrand fig 1.8 - G")
4894
add_to_lists(chartrand_18_g, graph_objects, all_graphs)
4895
4896
"""
4897
Chartrand Fig 1.8 - F1
4898
4899
sage: chartrand_18_f1.order()
4900
7
4901
sage: chartrand_18_f1.size()
4902
10
4903
"""
4904
chartrand_18_f1 = Graph("F@J]o")
4905
chartrand_18_f1.name(new = "chartrand fig 1.8 - F1")
4906
add_to_lists(chartrand_18_f1, graph_objects, all_graphs)
4907
4908
"""
4909
Chartrand Fig 1.8 - F2
4910
4911
sage: chartrand_18_f2.order()
4912
7
4913
sage: chartrand_18_f2.size()
4914
10
4915
"""
4916
chartrand_18_f2 = Graph("F?NVo")
4917
chartrand_18_f2.name(new = "chartrand fig 1.8 - F2")
4918
add_to_lists(chartrand_18_f2, graph_objects, all_graphs)
4919
4920
"""
4921
Chartrand Fig 1.10 - H1
4922
CE to independence_number(x) <= maximum(girth(x), card_center(x) + card_periphery(x))
4923
4924
sage: chartrand_110_h1.order()
4925
7
4926
sage: chartrand_110_h1.size()
4927
7
4928
"""
4929
chartrand_110_h1 = Graph("F@@Kw")
4930
chartrand_110_h1.name(new = "chartrand fig 1.10 - H1")
4931
add_to_lists(chartrand_110_h1, graph_objects, counter_examples, all_graphs)
4932
4933
"""
4934
Chartrand Fig 1.10 - H2
4935
4936
sage: chartrand_110_h2.order()
4937
7
4938
sage: chartrand_110_h2.size()
4939
7
4940
"""
4941
chartrand_110_h2 = Graph("F?C^G")
4942
chartrand_110_h2.name(new = "chartrand fig 1.10 - H2")
4943
add_to_lists(chartrand_110_h2, graph_objects, all_graphs)
4944
4945
"""
4946
Chartrand Fig 1.10 - H3
4947
4948
sage: chartrand_110_h3.order()
4949
9
4950
sage: chartrand_110_h3.size()
4951
8
4952
"""
4953
chartrand_110_h3 = Graph("H???]Os")
4954
chartrand_110_h3.name(new = "chartrand fig 1.10 - H3")
4955
add_to_lists(chartrand_110_h3, graph_objects, all_graphs)
4956
4957
"""
4958
Chartrand Fig 1.10 - H4
4959
4960
sage: chartrand_110_h4.order()
4961
9
4962
sage: chartrand_110_h4.size()
4963
8
4964
"""
4965
chartrand_110_h4 = Graph("H?C?JEK")
4966
chartrand_110_h4.name(new = "chartrand fig 1.10 - H4")
4967
add_to_lists(chartrand_110_h4, graph_objects, all_graphs)
4968
4969
# From their Mathing Theory book
4970
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')
4971
lovasz_plummer.name(new = "lovasz_plummer graph")
4972
add_to_lists(lovasz_plummer, graph_objects, all_graphs)
4973
4974
# Jorgenson graphs from: Jørgensen, Leif K. "Diameters of cubic graphs." Discrete applied mathematics 37 (1992): 347-351.
4975
jorgenson_1 = Graph('Ss??GOG?OC?_?I?DGCa?oDO?i??_C@?AC')
4976
jorgenson_1.name(new = "jorgenson_1")
4977
add_to_lists(jorgenson_1, graph_objects, all_graphs)
4978
4979
jorgenson_2 = Graph('GtP@Ww')
4980
jorgenson_2.name(new = "jorgenson_2")
4981
add_to_lists(jorgenson_2, graph_objects, all_graphs)
4982
4983
# CGT graphs come from
4984
# Examples and Counter Examples in Graph Theory by Michael Capobianco and John C. Molluzzo
4985
4986
cgt_5 = Graph("J??GkPPgbG?")
4987
cgt_5.name(new = "CGT Page 5")
4988
add_to_lists(cgt_5, graph_objects, all_graphs)
4989
4990
cgt_10_left = Graph("E`~o")
4991
cgt_10_left.name(new = "CGT Page 10, left")
4992
add_to_lists(cgt_10_left, graph_objects, all_graphs)
4993
4994
cgt_10_right = Graph("EFxw")
4995
cgt_10_right.name(new = "CGT Page 10, right")
4996
add_to_lists(cgt_10_right, graph_objects, all_graphs)
4997
4998
cgt_11_1221 = Graph("H`_YPN~")
4999
cgt_11_1221.name(new = "CGT Page 11, Figure 1.22.1")
5000
add_to_lists(cgt_11_1221, graph_objects, all_graphs)
5001
5002
cgt_13_bottom = Graph("Iv?GOKFY?")
5003
cgt_13_bottom.name(new = "CGT Page 13, Bottom")
5004
add_to_lists(cgt_13_bottom, graph_objects, all_graphs)
5005
5006
cgt_15_left = Graph("FD^vW")
5007
cgt_15_left.name(new = "CGT Page 15, left")
5008
add_to_lists(cgt_15_left, graph_objects, all_graphs)
5009
5010
# Sabidussi Graphs from Sabidussi, Gert. "The centrality index of a graph." Psychometrika 31.4 (1966): 581-603.
5011
5012
sabidussi_2 = Graph('Ls_aGcKSpiDCQH')
5013
sabidussi_2.name(new = "sabidussi_2")
5014
add_to_lists(sabidussi_2, graph_objects, all_graphs)
5015
5016
sabidussi_3 = Graph('H`?NeW{')
5017
sabidussi_3.name(new = "sabidussi_3")
5018
add_to_lists(sabidussi_3, graph_objects, all_graphs)
5019
5020
sabidussi_4 = Graph('L?O_?C@?G_oOF?')
5021
sabidussi_4.name(new = "sabidussi_4")
5022
add_to_lists(sabidussi_4, graph_objects, all_graphs)
5023
5024
sabidussi_5 = Graph('L??GH?GOC@AWKC')
5025
sabidussi_5.name(new = "sabidussi_5")
5026
add_to_lists(sabidussi_5, graph_objects, all_graphs)
5027
5028
sabidussi_6 = Graph('L??G?COGQCOCN?')
5029
sabidussi_6.name(new = "sabidussi_6")
5030
add_to_lists(sabidussi_6, graph_objects, all_graphs)
5031
5032
barnette = Graph("Ss?GOCDA?@_I@??_C?q?QC?_O@@??I??S")
5033
barnette.name(new = "Barnette-Bosak-Lederberg Graph")
5034
add_to_lists(barnette, graph_objects, all_graphs)
5035
5036
# Maximally Irregular Graphs
5037
mir_6 = MIR(6)
5038
mir_6.name(new = "max_irregular_6")
5039
add_to_lists(mir_6, graph_objects, all_graphs)
5040
5041
mir_7 = MIR(7)
5042
mir_7.name(new = "max_irregular_7")
5043
add_to_lists(mir_7, graph_objects, all_graphs)
5044
5045
# Radius Critical Ciliates
5046
c4_1 = Ciliate(2, 3)
5047
c4_1.name(new = "Ciliate 4, 1")
5048
add_to_lists(c4_1, graph_objects, all_graphs)
5049
5050
c4_2 = Ciliate(2, 4)
5051
c4_2.name(new = "Ciliate 4, 2")
5052
add_to_lists(c4_2, graph_objects, all_graphs)
5053
5054
c6_1 = Ciliate(3, 4)
5055
c6_1.name(new = "Ciliate 6, 1")
5056
add_to_lists(c6_1, graph_objects, all_graphs)
5057
5058
# Antiholes
5059
"""
5060
An odd antihole with 7 vertices
5061
p.175
5062
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.
5063
5064
sage: odd_antihole_7.order()
5065
7
5066
sage: odd_antihole_7.size()
5067
14
5068
"""
5069
antihole_7 = Antihole(7)
5070
antihole_7.name(new = "Antihole 7")
5071
add_to_lists(antihole_7, graph_objects, all_graphs)
5072
5073
antihole_8 = Antihole(8)
5074
antihole_8.name(new = "Antihole 8")
5075
add_to_lists(antihole_8, graph_objects, all_graphs)
5076
5077
"""
5078
p.10
5079
Barrus, Michael D. "On fractional realizations of graph degree sequences." arXiv preprint arXiv:1310.1112 (2013).
5080
"""
5081
fish = Graph("E@ro")
5082
fish.name(new = "fish")
5083
add_to_lists(fish, graph_objects, all_graphs)
5084
5085
"""
5086
p.10
5087
Barrus, Michael D. "On fractional realizations of graph degree sequences." arXiv preprint arXiv:1310.1112 (2013).
5088
"""
5089
fish_mod = Graph("EiKw")
5090
fish_mod.name(new = "fish_mod")
5091
add_to_lists(fish_mod, graph_objects, all_graphs)
5092
5093
"""
5094
All of the graphs with the name barrus_[0-9]{6} comes from the appendix of
5095
Barrus, Michael D. "On fractional realizations of graph degree sequences." arXiv preprint arXiv:1310.1112 (2013).
5096
"""
5097
5098
barrus_322111a = Graph("EAIW")
5099
barrus_322111a.name(new = "barrus_322111a")
5100
add_to_lists(barrus_322111a, graph_objects, all_graphs)
5101
5102
barrus_322111b = Graph("E?NO")
5103
barrus_322111b.name(new = "barrus_322111b")
5104
add_to_lists(barrus_322111b, graph_objects, all_graphs)
5105
5106
barrus_322221b = Graph("ECXo")
5107
barrus_322221b.name(new = "barrus_322221b")
5108
add_to_lists(barrus_322221b, graph_objects, all_graphs)
5109
5110
barrus_322221c = Graph("EAN_")
5111
barrus_322221c.name(new = "barrus_322221c")
5112
add_to_lists(barrus_322221c, graph_objects, all_graphs)
5113
5114
barrus_332211c = Graph("E@hW")
5115
barrus_332211c.name(new = "barrus_332211c")
5116
add_to_lists(barrus_332211c, graph_objects, all_graphs)
5117
5118
barrus_332211d = Graph("E?lo")
5119
barrus_332211d.name(new = "barrus_332211d")
5120
add_to_lists(barrus_332211d, graph_objects, all_graphs)
5121
5122
barrus_332222a = Graph("E`ow")
5123
barrus_332222a.name(new = "barrus_332222a")
5124
add_to_lists(barrus_332222a, graph_objects, all_graphs)
5125
5126
barrus_332222b = Graph("E`dg")
5127
barrus_332222b.name(new = "barrus_332222b")
5128
add_to_lists(barrus_332222b, graph_objects, all_graphs)
5129
5130
barrus_332222c = Graph("EoSw")
5131
barrus_332222c.name(new = "barrus_332222c")
5132
add_to_lists(barrus_332222c, graph_objects, all_graphs)
5133
5134
barrus_332222d = Graph("E_lo")
5135
barrus_332222d.name(new = "barrus_332222d")
5136
add_to_lists(barrus_332222d, graph_objects, all_graphs)
5137
5138
barrus_333221a = Graph("E`LW")
5139
barrus_333221a.name(new = "barrus_333221a")
5140
add_to_lists(barrus_333221a, graph_objects, all_graphs)
5141
5142
barrus_333221b = Graph("EKSw")
5143
barrus_333221b.name(new = "barrus_333221b")
5144
add_to_lists(barrus_333221b, graph_objects, all_graphs)
5145
5146
barrus_333221c = Graph("EELg")
5147
barrus_333221c.name(new = "barrus_333221c")
5148
add_to_lists(barrus_333221c, graph_objects, all_graphs)
5149
5150
barrus_333221d = Graph("EC\o")
5151
barrus_333221d.name(new = "barrus_333221d")
5152
add_to_lists(barrus_333221d, graph_objects, all_graphs)
5153
5154
barrus_333311 = Graph("E@lo")
5155
barrus_333311.name(new = "barrus_333311")
5156
add_to_lists(barrus_333311, graph_objects, all_graphs)
5157
5158
barrus_333322a = Graph("ES\o")
5159
barrus_333322a.name(new = "barrus_333322a")
5160
add_to_lists(barrus_333322a, graph_objects, all_graphs)
5161
5162
barrus_333322c = Graph("ED^_")
5163
barrus_333322c.name(new = "barrus_333322c")
5164
add_to_lists(barrus_333322c, graph_objects, all_graphs)
5165
5166
barrus_422211a = Graph("EGEw")
5167
barrus_422211a.name(new = "barrus_422211a")
5168
add_to_lists(barrus_422211a, graph_objects, all_graphs)
5169
5170
barrus_422211b = Graph("E?No")
5171
barrus_422211b.name(new = "barrus_422211b")
5172
add_to_lists(barrus_422211b, graph_objects, all_graphs)
5173
5174
barrus_432221a = Graph("E@pw")
5175
barrus_432221a.name(new = "barrus_432221a")
5176
add_to_lists(barrus_432221a, graph_objects, all_graphs)
5177
5178
barrus_432221b = Graph("E_Lw")
5179
barrus_432221b.name(new = "barrus_432221b")
5180
add_to_lists(barrus_432221b, graph_objects, all_graphs)
5181
5182
barrus_432221c = Graph("EANg")
5183
barrus_432221c.name(new = "barrus_432221c")
5184
add_to_lists(barrus_432221c, graph_objects, all_graphs)
5185
5186
barrus_432221d = Graph("E?^o")
5187
barrus_432221d.name(new = "barrus_432221d")
5188
add_to_lists(barrus_432221d, graph_objects, all_graphs)
5189
5190
barrus_433222a = Graph("E@vo")
5191
barrus_433222a.name(new = "barrus_433222a")
5192
add_to_lists(barrus_433222a, graph_objects, all_graphs)
5193
5194
barrus_433222c = Graph("EPVW")
5195
barrus_433222c.name(new = "barrus_433222c")
5196
add_to_lists(barrus_433222c, graph_objects, all_graphs)
5197
5198
barrus_433222d = Graph("E`NW")
5199
barrus_433222d.name(new = "barrus_433222d")
5200
add_to_lists(barrus_433222d, graph_objects, all_graphs)
5201
5202
barrus_433321a = Graph("EIMw")
5203
barrus_433321a.name(new = "barrus_433321a")
5204
add_to_lists(barrus_433321a, graph_objects, all_graphs)
5205
5206
barrus_433321b = Graph("E@^o")
5207
barrus_433321b.name(new = "barrus_433321b")
5208
add_to_lists(barrus_433321b, graph_objects, all_graphs)
5209
5210
barrus_433321c = Graph("EPTw")
5211
barrus_433321c.name(new = "barrus_433321c")
5212
add_to_lists(barrus_433321c, graph_objects, all_graphs)
5213
5214
barrus_433332a = Graph("EBzo")
5215
barrus_433332a.name(new = "barrus_433332a")
5216
add_to_lists(barrus_433332a, graph_objects, all_graphs)
5217
5218
barrus_433332b = Graph("E`^o")
5219
barrus_433332b.name(new = "barrus_433332b")
5220
add_to_lists(barrus_433332b, graph_objects, all_graphs)
5221
5222
barrus_433332c = Graph("EqLw")
5223
barrus_433332c.name(new = "barrus_433332c")
5224
add_to_lists(barrus_433332c, graph_objects, all_graphs)
5225
5226
barrus_442222a = Graph("E?~o")
5227
barrus_442222a.name(new = "barrus_442222a")
5228
add_to_lists(barrus_442222a, graph_objects, all_graphs)
5229
5230
barrus_442222b = Graph("E_lw")
5231
barrus_442222b.name(new = "barrus_442222b")
5232
add_to_lists(barrus_442222b, graph_objects, all_graphs)
5233
5234
barrus_443322a = Graph("E@~o")
5235
barrus_443322a.name(new = "barrus_443322a")
5236
add_to_lists(barrus_443322a, graph_objects, all_graphs)
5237
5238
barrus_443322b = Graph("EHuw")
5239
barrus_443322b.name(new = "barrus_443322b")
5240
add_to_lists(barrus_443322b, graph_objects, all_graphs)
5241
5242
barrus_443322c = Graph("EImw")
5243
barrus_443322c.name(new = "barrus_443322c")
5244
add_to_lists(barrus_443322c, graph_objects, all_graphs)
5245
5246
barrus_443322d = Graph("EQlw")
5247
barrus_443322d.name(new = "barrus_443322d")
5248
add_to_lists(barrus_443322d, graph_objects, all_graphs)
5249
5250
barrus_443322e = Graph("E`lw")
5251
barrus_443322e.name(new = "barrus_443322e")
5252
add_to_lists(barrus_443322e, graph_objects, all_graphs)
5253
5254
barrus_443331a = Graph("EBxw")
5255
barrus_443331a.name(new = "barrus_443331a")
5256
add_to_lists(barrus_443331a, graph_objects, all_graphs)
5257
5258
barrus_443331b = Graph("E`\w")
5259
barrus_443331b.name(new = "barrus_443331b")
5260
add_to_lists(barrus_443331b, graph_objects, all_graphs)
5261
5262
barrus_443333a = Graph("Es\w")
5263
barrus_443333a.name(new = "barrus_443333a")
5264
add_to_lists(barrus_443333a, graph_objects, all_graphs)
5265
5266
barrus_443333c = Graph("Eqlw")
5267
barrus_443333c.name(new = "barrus_443333c")
5268
add_to_lists(barrus_443333c, graph_objects, all_graphs)
5269
5270
barrus_444332b = Graph("Ed\w")
5271
barrus_444332b.name(new = "barrus_444332b")
5272
add_to_lists(barrus_444332b, graph_objects, all_graphs)
5273
5274
barrus_444332c = Graph("EMlw")
5275
barrus_444332c.name(new = "barrus_444332c")
5276
add_to_lists(barrus_444332c, graph_objects, all_graphs)
5277
5278
barrus_444433a = Graph("ER~o")
5279
barrus_444433a.name(new = "barrus_444433a")
5280
add_to_lists(barrus_444433a, graph_objects, all_graphs)
5281
5282
barrus_444433b = Graph("Et\w")
5283
barrus_444433b.name(new = "barrus_444433b")
5284
add_to_lists(barrus_444433b, graph_objects, all_graphs)
5285
5286
"""
5287
From:
5288
Alcón, Liliana, Marisa Gutierrez, and Glenn Hurlbert. "Pebbling in split graphs." SIAM Journal on Discrete Mathematics 28.3 (2014): 1449-1466.
5289
"""
5290
pyramid = Graph("EElw")
5291
pyramid.name(new = "pyramid")
5292
add_to_lists(pyramid, graph_objects, all_graphs)
5293
5294
"""
5295
From:
5296
p.3
5297
Dvořák, Zdeněk, and Jordan Venters. "Triangle-free planar graphs with small independence number." arXiv preprint arXiv:1702.02888 (2017).
5298
"""
5299
thomas_walls_3 = Graph("IRaIACbFG")
5300
thomas_walls_3.name(new = "thomas_walls_3")
5301
add_to_lists(thomas_walls_3, graph_objects, all_graphs)
5302
5303
"""
5304
From:
5305
p.3
5306
Dvořák, Zdeněk, and Jordan Venters. "Triangle-free planar graphs with small independence number." arXiv preprint arXiv:1702.02888 (2017).
5307
"""
5308
thomas_walls_4 = Graph("LQCkCD?OGM@EKB")
5309
thomas_walls_4.name(new = "thomas_walls_4")
5310
add_to_lists(thomas_walls_4, graph_objects, all_graphs)
5311
5312
"""
5313
P. 24, 1st figure
5314
From a knuth paper, linked in issue #342
5315
"""
5316
knuth_24_1 = Graph("E`~w")
5317
knuth_24_1.name(new = "knuth_24_1")
5318
add_to_lists(knuth_24_1, graph_objects, all_graphs)
5319
5320
"""
5321
P. 24, 2nd figure
5322
From a knuth paper, linked in issue #343
5323
"""
5324
knuth_24_2 = Graph("Er~w")
5325
knuth_24_2.name(new = "knuth_24_2")
5326
add_to_lists(knuth_24_2, graph_objects, all_graphs)
5327
5328
"""
5329
From Craig Larson, Critical Independence paper
5330
Has non empty critical independence set
5331
"""
5332
larson = Graph("F@N~w")
5333
larson.name(new = "larson")
5334
add_to_lists(larson, graph_objects, all_graphs)
5335
5336
"""
5337
From:
5338
Vizing's independence number conjecture is true asymptotically
5339
Eckhard Steffen
5340
"""
5341
steffen_1 = Graph("HqP@xw{")
5342
steffen_1.name(new = "steffen_1")
5343
add_to_lists(steffen_1, graph_objects, all_graphs)
5344
5345
"""
5346
From:
5347
Vizing's independence number conjecture is true asymptotically
5348
Eckhard Steffen
5349
"""
5350
steffen_2 = Graph("N??xpow@o@?AoBoBW@_")
5351
steffen_2.name(new = "steffen_2")
5352
add_to_lists(steffen_2, graph_objects, all_graphs)
5353
5354
"""
5355
From:
5356
Vizing's independence number conjecture is true asymptotically
5357
Eckhard Steffen
5358
"""
5359
steffen_3 = Graph("Z??xpow@o@?A?B?B?@_?C??_?@??@??A???__?Dc??kO?AoO?AWO?AWG?@K?")
5360
steffen_3.name(new = "steffen_3")
5361
add_to_lists(steffen_3, graph_objects, all_graphs)
5362
5363
"""
5364
From:
5365
p.231
5366
Balinski, Michel L. "Notes—On a Selection Problem." Management Science 17.3 (1970): 230-231.
5367
"""
5368
balanski = Graph("H??xuRo")
5369
balanski.name(new = "balanski")
5370
add_to_lists(balanski, graph_objects, all_graphs)
5371
5372
"""
5373
Instance of the Erdos-Faber-Lovasz conjecture
5374
"""
5375
efl_instance = Graph("J`?GECrB~z_")
5376
efl_instance.name(new = "efl_instance")
5377
add_to_lists(efl_instance, graph_objects, all_graphs)
5378
5379
"""
5380
From Lipták, László, and László Lovász. "Critical facets of the stable set polytope." Combinatorica 21.1 (2001): 61-88.
5381
with defect 2
5382
"""
5383
basic_facet_1 = Graph("FCS~?")
5384
basic_facet_1.name(new = "basic facet-graph")
5385
add_to_lists(basic_facet_1, graph_objects, all_graphs)
5386
5387
"""
5388
From Lipták, László, and László Lovász. "Critical facets of the stable set polytope." Combinatorica 21.1 (2001): 61-88.
5389
with defect 2
5390
"""
5391
basic_facet_2 = Graph("I_?BSgcEG")
5392
basic_facet_2.name(new = "basic facet-graph two")
5393
add_to_lists(basic_facet_2, graph_objects, all_graphs)
5394
5395
"""
5396
From Lipták, László, and László Lovász. "Critical facets of the stable set polytope." Combinatorica 21.1 (2001): 61-88.
5397
with defect 2
5398
"""
5399
basic_facet_3 = Graph("L@Q????o`QAc?w")
5400
basic_facet_3.name(new = "basic facet-graph three")
5401
add_to_lists(basic_facet_3, graph_objects, all_graphs)
5402
5403
"""
5404
From Lipták, László, and László Lovász. "Critical facets of the stable set polytope." Combinatorica 21.1 (2001): 61-88.
5405
"""
5406
crit_facet_1 = Graph("H@BDQo{")
5407
crit_facet_1.name(new = "critical facet graph 1")
5408
add_to_lists(crit_facet_1, graph_objects, all_graphs)
5409
5410
"""
5411
From Lipták, László, and László Lovász. "Critical facets of the stable set polytope." Combinatorica 21.1 (2001): 61-88.
5412
"""
5413
crit_facet_2 = Graph("J?AAHGIC^o?")
5414
crit_facet_2.name(new = "critical facet graph 2")
5415
add_to_lists(crit_facet_2, graph_objects, all_graphs)
5416
5417
"""
5418
From:
5419
p.222
5420
Tarjan, Robert E. "Decomposition by clique separators." Discrete mathematics 55.2 (1985): 221-232.
5421
"""
5422
tarjan = Graph("JQ?CQOKDXL_")
5423
tarjan.name(new = "tarjan")
5424
add_to_lists(tarjan, graph_objects, all_graphs)
5425
5426
"""
5427
From:
5428
p.225
5429
Tarjan, Robert E. "Decomposition by clique separators." Discrete mathematics 55.2 (1985): 221-232.
5430
"""
5431
tarjan_fillin = Graph("JQAa_OFJWv_")
5432
tarjan_fillin.name(new = "tarjan_fillin")
5433
add_to_lists(tarjan_fillin, graph_objects, all_graphs)
5434
5435
"""
5436
From:
5437
p.252
5438
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.
5439
"""
5440
sewell_1 = Graph("H@GOkXI")
5441
sewell_1.name(new = "sewell_1")
5442
add_to_lists(sewell_1, graph_objects, all_graphs)
5443
5444
"""
5445
Graphs which are two cliques somehow connected to each other
5446
"""
5447
two_clique_1 = Graph("OwCW?CB???_Bw?F?_[W?~")
5448
two_clique_1.name(new = "two-clique 1")
5449
add_to_lists(two_clique_1, graph_objects, all_graphs)
5450
5451
two_clique_2 = Graph("FwC^w")
5452
two_clique_2.name(new = "two-clique 2")
5453
add_to_lists(two_clique_2, graph_objects, all_graphs)
5454
5455
caro_roditty_4 = Caro_Roditty(4)
5456
caro_roditty_4.name(new = "Caro-Roditty 4 Graph")
5457
add_to_lists(caro_roditty_4, graph_objects, all_graphs)
5458
5459
caro_roditty_5 = Caro_Roditty(5)
5460
caro_roditty_5.name(new = "Caro-Roditty 5 Graph")
5461
add_to_lists(caro_roditty_5, graph_objects, all_graphs)
5462
5463
jakovac_2a = Graph("a???????GO???@???_?????C??W????????@_A???B??W??[??????X?E?Q?????????C??Bb???C?o?K??s?@?GZ??F?Ew")
5464
jakovac_2a.name(new = "jakovac fig 2a")
5465
add_to_lists(jakovac_2a, graph_objects, all_graphs)
5466
5467
jakovac_2b = Graph("a???????????????????????@???C??????C?????oO??_I?G?o@_??[????o?O`?????????G???AW???_[??O_N??W?Bw")
5468
jakovac_2b.name(new = "jakovac fig 2b")
5469
add_to_lists(jakovac_2b, graph_objects, all_graphs)
5470
5471
jakovac_5 = Graph("Q`?G?C??G??@?B?K?W?W@o?go@O")
5472
jakovac_5.name(new = "jakovac fig 5")
5473
add_to_lists(jakovac_5, graph_objects, all_graphs)
5474
5475
# From A survey of selected recent results on total domination in graphs By Michael A. Henning
5476
henning_5 = Graph("H?qipno")
5477
henning_5.name(new = "henning fig 5")
5478
add_to_lists(henning_5, graph_objects, all_graphs)
5479
5480
henning_8 = Graph("[SP???_C?O?_?_?O?G?C????????CO?AG??c??E??W??E?????@???A_??QG??BD")
5481
henning_8.name(new = "henning fig 8")
5482
add_to_lists(henning_8, graph_objects, all_graphs)
5483
5484
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")
5485
henning_13.name(new = "henning fig 13")
5486
add_to_lists(henning_13, graph_objects, all_graphs)
5487
5488
henning_14 = Graph("M_GSA?_C?G@??b?[?")
5489
henning_14.name(new = "henning fig 14")
5490
add_to_lists(henning_14, graph_objects, all_graphs)
5491
5492
henning_15 = graphs.CycleGraph(4)
5493
for v in henning_15.vertices():
5494
count = len(henning_15.vertices())
5495
henning_15.add_cycle([count..count+9])
5496
henning_15.add_edge((v, count))
5497
henning_15.name(new = "henning fig 15")
5498
add_to_lists(henning_15, graph_objects, all_graphs)
5499
5500
henning_16_g1 = Graph("Fq?gw")
5501
henning_16_g1.name(new = "henning fig 16 g1")
5502
add_to_lists(henning_16_g1, graph_objects, all_graphs)
5503
5504
henning_16_g2 = Graph("FWEYo")
5505
henning_16_g2.name(new = "henning fig 16 g2")
5506
add_to_lists(henning_16_g2, graph_objects, all_graphs)
5507
5508
henning_16_g3 = Graph("JQQ@?cCAGF_")
5509
henning_16_g3.name(new = "henning fig 16 g3")
5510
add_to_lists(henning_16_g3, graph_objects, all_graphs)
5511
5512
henning_16_g4 = Graph("NC`@A?_C?@_JA??___W")
5513
henning_16_g4.name(new = "henning fig 16 g4")
5514
add_to_lists(henning_16_g4, graph_objects, all_graphs)
5515
5516
henning_16_g5 = Graph("JSP@?cG@GF_")
5517
henning_16_g5.name(new = "henning fig 16 g5")
5518
add_to_lists(henning_16_g5, graph_objects, all_graphs)
5519
5520
henning_16_g6 = Graph("LSP??CAGOa`C@H")
5521
henning_16_g6.name(new = "henning fig 16 g6")
5522
add_to_lists(henning_16_g6, graph_objects, all_graphs)
5523
5524
henning_16_g7 = Graph("NCGaC@?C?@_J?_A?__W")
5525
henning_16_g7.name(new = "henning fig 16 g7")
5526
add_to_lists(henning_16_g7, graph_objects, all_graphs)
5527
5528
"""
5529
From:
5530
Weighted and Unweighted Maximum Clique Algorithms with Upper Bounds from Fractional Coloring
5531
E. Balas and Jue Xue
5532
"""
5533
balas_1 = Graph("HWFZvq}")
5534
balas_1.name(new = "balas_1")
5535
add_to_lists(balas_1, graph_objects, all_graphs)
5536
5537
"""
5538
From:
5539
On alpha+ - Stable K¨onig-Egervary Graphs
5540
Vadim E. Levit and Eugen Mandrescu
5541
"""
5542
levit_1 = Graph("G?rHx{")
5543
levit_1.name(new = "levit_1")
5544
add_to_lists(levit_1, graph_objects, all_graphs)
5545
5546
"""
5547
From:
5548
On alpha+ - Stable K¨onig-Egervary Graphs
5549
Vadim E. Levit and Eugen Mandrescu
5550
"""
5551
levit_2 = Graph("GCOihs")
5552
levit_2.name(new = "levit_2")
5553
add_to_lists(levit_2, graph_objects, all_graphs)
5554
5555
"""
5556
From:
5557
On alpha+ - Stable K¨onig-Egervary Graphs
5558
Vadim E. Levit and Eugen Mandrescu
5559
"""
5560
levit_3 = Graph("G@G]@{")
5561
levit_3.name(new = "levit_3")
5562
add_to_lists(levit_3, graph_objects, all_graphs)
5563
5564
"""
5565
From:
5566
On alpha-Critical Edges in K¨onig-Egerv´ary Graphs
5567
Vadim E. Levit and Eugen Mandrescu
5568
"""
5569
levit_4 = Graph("KGD?_O_@GE~~")
5570
levit_4.name(new = "levit_4")
5571
add_to_lists(levit_4, graph_objects, all_graphs)
5572
5573
5574
5575
#graph_objects: all graphs with no duplicates
5576
#obvious way to remove duplicates in list of ALL objects
5577
5578
"""
5579
graph_objects = []
5580
for g in union_objects, idfun=Graph.graph6_string:
5581
if not g in graph_objects:
5582
graph_objects.append(g)
5583
"""
5584
5585
#fast way to remove duplicates in list of ALL objects
5586
#from : http://www.peterbe.com/plog/uniqifiers-benchmark
5587
5588
5589
def remove_duplicates(seq, idfun=None):
5590
# order preserving
5591
if idfun is None:
5592
def idfun(x): return x
5593
seen = {}
5594
result = []
5595
for item in seq:
5596
marker = idfun(item)
5597
# in old Python versions:
5598
# if seen.has_key(marker)
5599
# but in new ones:
5600
if marker in seen: continue
5601
seen[marker] = 1
5602
result.append(item)
5603
return result
5604
5605
#could run this occasionally to check there are no duplicates
5606
#graph_objects = remove_duplicates(union_objects, idfun=Graph.graph6_string)
5607
# Class 0 pebbling graphs
5608
5609
class0graphs_dict = {
5610
'C05': 'IheA@GUAo',
5611
'C05C05': 'XheAHCPBGG?P?P?G_BG?O?@C?AG?AG?@e??OO?AH??Ga??PA??X',
5612
'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',
5613
'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_',
5614
'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',
5615
'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',
5616
'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',
5617
'C05K02': 'IrGWOMAOW',
5618
'C05K03': 'N{Sw_SF?_A_F_COAc?w',
5619
'C05K04': 'S~`HW{G@GD_N?G?C_@W?N_?`?@H?@W_?{',
5620
'C05K05': 'X~}AHKVBwG?P?R?J_Bw?O?@C?AW?Aw?@}??OO?AH??Ha??VA??^',
5621
'C05K06': ']~~{ACbCwV_~?_?O_CW?f?A{?Fw?C??AC??b??Cw??V_??~_??_O??Oc??CW_??fA??A{C??Fw',
5622
'C05K07': 'b~~~{@@GWb`N@^?~_G?@@?CK?Gw?Hw?D{?@~??G???__?@B??@F???f_??Jw??@~_??C@???GH???GW_??C[G??@N@???JwC???~_',
5623
'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~',
5624
'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~',
5625
'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~_',
5626
'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',
5627
'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~_?_?????~~',
5628
'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~{',
5629
'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',
5630
'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',
5631
'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??_???????~~{',
5632
'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????????^~~',
5633
'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',
5634
'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?????????~~~_',
5635
'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~~~',
5636
'C05Q2': 'Sr`HOkG@GD?J?G?C_@O?J_?`?@H?@O_?k',
5637
'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',
5638
'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',
5639
'C06Pow02': 'EznW',
5640
'C07Pow02': 'FzM]W',
5641
'C08Pow02': 'GzK[]K',
5642
'C09Pow02': 'HzKW[NB',
5643
'C10Pow04': 'I~||}~n|w',
5644
'C11Pow04': 'J~|x{~Nx~f_',
5645
'C12Pow04': 'K~|xw}Np~F}N',
5646
'C12Pow05': 'K~~z|~^z~n~^',
5647
'C13Pow04': 'L~|xw{N`}F{N{N',
5648
'C14Pow04': 'M~|xw{N@{FwNwN{F_',
5649
'C15Pow04': 'N~|xw{N@wFoNoNwF}@w',
5650
'C16Pow04': 'O~|xw{N@wF_N_NoF{@~_N',
5651
'C17Pow04': 'P~|xw{N@wF_N?N_Fw@~?N{?{',
5652
'C18Pow04': 'Q~|xw{N@wF_N?N?Fo@}?Nw?~o@w',
5653
'C19Pow04': 'R~|xw{N@wF_N?N?F_@{?No?~_@~_@w',
5654
'C20Pow04': 'S~|xw{N@wF_N?N?F_@w?N_?~?@~?@~_?{',
5655
'C21Pow04': 'T~|xw{N@wF_N?N?F_@w?N??}?@}?@~??~o?N',
5656
'C22Pow04': 'U~|xw{N@wF_N?N?F_@w?N??{?@{?@}??~_?N{?@w',
5657
'C23Pow04': 'V~|xw{N@wF_N?N?F_@w?N??{?@w?@{??~??Nw?@~_?F_',
5658
'C24Pow08': 'W~~~~~~^x~b~B~@~_^wB~_N~?^~?^~_N~wB~~?^~{@~~wB~',
5659
'C25Pow08': 'X~~~~~~^x~b~B~@~_^wB~?N}?^}?^~?N~oB~}?^~w@~~oB~~oB~',
5660
'C26Pow08': 'Y~~~~~~^x~b~B~@~_^wB~?N{?^{?^}?N~_B~{?^~o@~~_B~~_B~~o@~_',
5661
'C27Pow08': 'Z~~~~~~^x~b~B~@~_^wB~?N{?^w?^{?N~?B~w?^~_@~~?B~~?B~~_@~~w?^w',
5662
'C28Pow08': '[~~~~~~^x~b~B~@~_^wB~?N{?^w?^w?N}?B~o?^~?@~}?B~}?B~~?@~~o?^~}?B~',
5663
'C29Pow08': '\\~~~~~~^x~b~B~@~_^wB~?N{?^w?^w?N{?B~_?^}?@~{?B~{?B~}?@~~_?^~{?B~~o?N{',
5664
'C30Pow08': ']~~~~~~^x~b~B~@~_^wB~?N{?^w?^w?N{?B~??^{?@~w?B~w?B~{?@~~??^~w?B~~_?N~~??^w',
5665
'C31Pow08': '^~~~~~~^x~b~B~@~_^wB~?N{?^w?^w?N{?B~??^w?@~o?B~o?B~w?@~}??^~o?B~~??N~}??^~}??^w',
5666
'C32Pow08': '_~~~~~~^x~b~B~@~_^wB~?N{?^w?^w?N{?B~??^w?@~_?B~_?B~o?@~{??^~_?B~}??N~{??^~{??^~}??N{',
5667
'C33Pow08': '`~~~~~~^x~b~B~@~_^wB~?N{?^w?^w?N{?B~??^w?@~_?B~??B~_?@~w??^~??B~{??N~w??^~w??^~{??N~~??B~',
5668
'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',
5669
'C35Pow08': 'b~~~~~~^x~b~B~@~_^wB~?N{?^w?^w?N{?B~??^w?@~_?B~??B~??@~_??^{??B~o??N~_??^~_??^~o??N~{??B~~_??^~}??@~_',
5670
'C36Pow08': 'c~~~~~~^x~b~B~@~_^wB~?N{?^w?^w?N{?B~??^w?@~_?B~??B~??@~_??^w??B~_??N~???^~???^~_??N~w??B~~???^~{??@~~w??B~',
5671
'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~',
5672
'C38Pow08': 'e~~~~~~^x~b~B~@~_^wB~?N{?^w?^w?N{?B~??^w?@~_?B~??B~??@~_??^w??B~???N{???^{???^}???N~_??B~{???^~o??@~~_??B~~_??B~~o??@~_',
5673
'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',
5674
'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~',
5675
'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{',
5676
'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',
5677
'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',
5678
'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{',
5679
'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~',
5680
'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',
5681
'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~~_????^~}????@~_',
5682
'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~',
5683
'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~',
5684
'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????@~_',
5685
'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',
5686
'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~',
5687
'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{',
5688
'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',
5689
'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',
5690
'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{',
5691
'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~',
5692
'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',
5693
'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~~_??????^~}??????@~_',
5694
'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~~',
5695
'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~~',
5696
'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~~_',
5697
'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',
5698
'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~~',
5699
'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~~~~{?????~~{',
5700
'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',
5701
'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',
5702
'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~~~~_?????~~~~~?????@~~~~~?????@~~~~~_?????~~{',
5703
'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~~',
5704
'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',
5705
'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~~_',
5706
'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~~',
5707
'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~~',
5708
'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~~_',
5709
'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',
5710
'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~~',
5711
'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~~~~{???????~~{',
5712
'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',
5713
'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',
5714
'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~~~~_???????~~~~~???????@~~~~~???????@~~~~~_???????~~{',
5715
'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~~',
5716
'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',
5717
'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~~_',
5718
'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~~',
5719
'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~~',
5720
'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~~_',
5721
'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',
5722
'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~~',
5723
'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~~~~{?????????~~{',
5724
'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',
5725
'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',
5726
'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~~~~_?????????~~~~~?????????@~~~~~?????????@~~~~~_?????????~~{',
5727
'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~~',
5728
'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',
5729
'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~~_',
5730
'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~~',
5731
'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~~',
5732
'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~~_',
5733
'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',
5734
'DenBip04': 'G?vtb_',
5735
'DenBip05': 'I?B^fPwm?',
5736
'DenBip06': 'K??BjzgnFw]?',
5737
'DenBip07': 'M???FfqNf{\\of_~??',
5738
'DenBip08': 'O????Bl~BM]Ohou_ZGC}?',
5739
'DenBip09': 'Q??????~m}TknozwVKBf?]gBe_?',
5740
'DenBip10': 'S???????F}{[v_rKFwDJ_mKBPo@^_Bl??',
5741
'DenBip11': 'U?????????H}v][u^rA^OjeBMgFf?Dz_@sw?m[??',
5742
'DenBip12': 'W???????????}mh]UReyGJZBG{FuGBiW@fo?fv?E~o?E}??',
5743
'DenBip13': 'Y?????????????\\{u{q^gznb[uDw{FYS@V[?I~?Fqc?]XO?}J_?bn???',
5744
'DenBip14': '[???????????????\\l}Lk^sOt]DW{@\\iBVL?YVoE|C?J\\G?kho?k}??SZg?Er]??',
5745
'DenBip15': ']?????????????????@bv{Hzvgdy~E\\mBJr_JZgBx|?P|o?xUo?Rv_?Uww?A}~??mnW?BY{???',
5746
'DenBip16': '_????????????????????s]V\\ndXs`rR`YmO}fOC}T?Jy]?wnC?jHw?VmG?Am|??{|{?Bng_?FM~_?F@^_??',
5747
'DenBip17': 'a??????????????????????@}k~^JtUdoTzW|hKEp^?E}Y?zRy?F\\q?^~N?Drp_?Hj{?BiVG?EUko?Apz_?BL^G??S~e???',
5748
'DenBip18': 'c?????????????????????????FZ{^dqQ^foW}iDkTgU|__]z]?om]?PP|_Ahuo?LMm?B|Vo?DYmG?EyzW?BkBk??Ufd??DFhg??H]F_??',
5749
'DenBip19': 'e????????????????????????????B_v|DlmlRsfF\\cFNx?y\\F_va~?RZkODWNk?ulh??vsk?D^_k?@uFw?@unE??T}c_?FJbo??X~No??vlR_??nIf_???',
5750
'DenBip20': 'g???????????????????????????????Bw[s}Y`ufbYUt\\oUF{_glyOTwsWAjTq?EVfOAYZe?FuLm??^S}?Bv|Q??plx??FXyc??@uZg??Z^R???lLN_??Kw\\W??@Avy???',
5751
'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_???',
5752
'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_???',
5753
'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???~[~}????',
5754
'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_???',
5755
'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_????',
5756
'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}????',
5757
'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_????',
5758
'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_????',
5759
'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?????',
5760
'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_?????',
5761
'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`??????',
5762
'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?????',
5763
'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??????',
5764
'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??????',
5765
'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??????',
5766
'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_?????',
5767
'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???????',
5768
'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??????',
5769
'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???????',
5770
'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_??????',
5771
'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???????',
5772
'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|???????',
5773
'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_???????',
5774
'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???????',
5775
'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_???????',
5776
'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`_???????',
5777
'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????????',
5778
'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_???????',
5779
'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_????????',
5780
'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_[????????',
5781
'K02': 'A_',
5782
'K02K03': 'E{Sw',
5783
'K02K04': 'G~`HW{',
5784
'K02K05': 'I~}AHKVBw',
5785
'K02K06': 'K~~{ACbCwV_~',
5786
'K02K07': 'M~~~{@@GWb`N@^?~_',
5787
'K02K08': 'O~~~~}?OH@aFAN@N_VwB~',
5788
'K02K09': 'Q~~~~~~_A?cBCFAF_bwC~?V{?~w',
5789
'K02K10': 'S~~~~~~~{?G@GBCB`@wG^?b{@Nw@^w?~{',
5790
'K02K11': 'U~~~~~~~~~o?O@G@a?wON@@{AFwANw@N{?V~?B~w',
5791
'K02K12': 'W~~~~~~~~~~~_?O?c?W_FA?{CBwCFwAF{?b~?C~w?V~_?~~',
5792
'K02K13': 'Y~~~~~~~~~~~~~_?G?H?BC?[G@wGBwCB{@@~?G^w?b~_@N~?@^~??~~_',
5793
'K02K14': '[~~~~~~~~~~~~~~~o?A?@G?KO?wO@wG@{A?~?ONw@@~_AF~?AN~?@N~_?V~w?B~~',
5794
'K02K15': ']~~~~~~~~~~~~~~~~~{??O?C_?W_?wO?{C?^?_FwA?~_CB~?CF~?AF~_?b~w?C~~??V~{??~~w',
5795
'K02K16': '_~~~~~~~~~~~~~~~~~~~~_?@??H??W_?[G?N@?BwC?^_G@~?GB~?CB~_@@~w?G^~??b~{?@N~w?@^~w??~~{',
5796
'K02K17': 'a~~~~~~~~~~~~~~~~~~~~~~}??A??H??KO?FA?@wG?N_O?~?O@~?G@~_A?~w?ON~?@@~{?AF~w?AN~w?@N~{??V~~??B~~w',
5797
'K02K18': 'c~~~~~~~~~~~~~~~~~~~~~~~~~{??A??C_?BC??wO?F__?^?_?~?O?~_C?^w?_F~?A?~{?CB~w?CF~w?AF~{??b~~??C~~w??V~~_??~~~',
5798
'K02K19': 'e~~~~~~~~~~~~~~~~~~~~~~~~~~~~{??@??@G??W_?B`??N@??^?_?^_G?Nw@?B~?C?^{?G@~w?GB~w?CB~{?@@~~??G^~w??b~~_?@N~~??@^~~???~~~_',
5799
'K02K20': 'g~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}???O??H??@a??FA??N@??N_O?FwA?@~?G?N{?O?~w?O@~w?G@~{?A?~~??ON~w?@@~~_?AF~~??AN~~??@N~~_??V~~w??B~~~',
5800
'K03': 'Bw',
5801
'K03K03': 'H{S{aSf',
5802
'K03K03K03': 'Z{S{aSf_A?cBCCAA__wCc?QS?c{?_A?OC_CBC?__OAA__CF?_Cc?OAQ_C?cw',
5803
'K03K03K04': 'c~`HW}GPHDaN_?O?c?W_FA?_C@GC@WA?{?aG?CPG?PD_?aN_?_?O?O?c?C?W_?_FA?A?_C?C@GC?C@WA?A?{?_?aG?C?CPG?O?PD_?_?aN',
5804
'K03K03K05': 'l~}AHKVB{GGPGRCJ`B{??O?C_?W_?wO?{C?O?_AGA?H_C?V?C?^?ACG??`AG?CGR??O`[??`B{??_?A??O?C_?C?BC??_?wO?A?F__?C?O?_?C?P?O?A?H_C??_Aw?_?C?^?A??O`??C??`AG?C??`AW?A??O`[??_?CG^',
5805
'K03K03K06': 'u~~{ACbCwV_~__OOcCW_fAA{CF{??A??C_?BC??wO?F__?^?_?_?O?O_C?CW?_?f?A?A{?C?Fw?CCC??AAAC??__b??CCCw??OOV_??__~_??_??O??O??c??C??W_??_?FA??A??{C??C?BwC??C?C?A??A?AC?_??_?b?C??C?Cw?O??O?V_?_??_?~??_??___??O??OOO_?C??CCCW??_??__f??A??AAA{??C??CCFw',
5806
'K03K03K07': '~??~~~~{@@GWb`N@^?~oGA@@GCKOGwOHwGD{A@~_??A???c??BC??FA??F__?BwC??~?O?C??_?GG?_?GW?O?C[?C?@N??_?Jw?A??~_?CA@???CA@@??A@?`_??_OGw??CA@N???OGD{???_ON{???_??A???O??C_??C??BC???_??wO??A??F__??C??^?_??C??~?O??A??_?C???_?GG?_??C?@B?A???O?C[?C???_?Hw?C???_?Jw?A???O?F{??_??CA@???C???_OGG??O??A@?`_??_??CA@F???_??CA@N???O??A@?n_??C???_ONw',
5807
'K03K03K08': '~?@G~~~~}?OH@aFAN@N_VwB~_G@?OH?OW_G[GAN@?RwC@^_GB~_???O???c???W_??FA???{C??BwC??FwA??F{?_?A??C??OG?O?@@_?_?AF??_?AN??O?@N_?C??Vw??_?B~??A?_G???C@?OG??C@?OW??A?_G[???_GAN???C@?Rw???OC@^_???_GB~_???_???O???O???c???C???W_???_??FA???A???{C???C??BwC???C??FwA???A??F{?_???_?A??C???C??OG?O???O?@@_?_???_?AF??_???_?AN??O???O?@N_?C???C??Vw??_???_?B~??A???A?_G???C???C@?OG??C???C@?OW??A???A?_G[???_???_GAN???C???C@?Rw???O???OC@^_???_???_GB~',
5808
'K03K03K09': '~?@P~~~~~~_A?cBCFAF_bwC~?V{?~{?_A?OC_CBC?_wOAF__C^?_C~?OA~_C?~{????O???C_???W_???wO???{C???^?_??FwA???~_C??B~?C??C??A??A?_?_??_W?C??CF??O??O{??_??bw??_??fw??O??V{??C??F~???_C?_???A?OA?_??C?_CB???C?_CF???A?OAF_???_C?bw???C?_C~????OA?V{????_C?~{????_???A????O???C_???C???BC????_???wO???A???F__???C???^?_???C???~?O???A???~_C????_??^w?_???C??C??A????O??OC?C????_??_W?C????_??_w?A????O??O{??_???C??C^??C????_??fw??O???A??A~_??_???C??F~???_???C?_C????O???A?OA?_??C????_C?_W???_???C?_CF???A????OA?O{???C????_C?bw???C????_C?fw???A????OA?V{????_???C?_F~',
5809
'K03K03K10': '~?@Y~~~~~~~{?G@GBCB`@wG^?b{@Nw@^w?~}?G?O@?H?C@a?GFA?GN@?CN_O@FwA?H~?G?n{?O@~{????A????C_???BC????wO???F__???^?_???~?O???~_C???^w?_??F~?A???_??C??@?G?C??@?W?A???_[??_??GN??C??@Bw??O??C^_??_??H~???_??J~???O??F~_??C?O@?????_A?G@???A?G?_K???C?O@?w???C?O@@w???A?G?`{????_A?G~????C?O@Nw????O@?D~_????_A?N~_????_????O????O????c????C????W_????_???FA????A????{C????C???BwC????C???FwA????A???F{?_????_??B~?C????C???~w?O????O??C???_????_??G@??_????_??GB??O????O??CB_?C????C??@@w??_????_??G^??A????A???b{??C????C??@Nw??C????C??@^w??A????A???~{???_????_A?G????C????C?O@?G???O????O@?C@_???_????_A?GF????_????_A?GN????O????O@?CN_???C????C?O@Fw????_????_A?H~????A????A?G?n{????C????C?O@~w',
5810
'K03K03K11': '~?@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?????c????BC????FA????F__???BwC????~?O???F{?_???^w?_???~w?O???~{?C???O???_??A?G?A???G@_?C???OF??C???ON??A???GN_??_??AFw??C???P~???O??@N{???_??A~w???_??B~w???O?_@?????C?G?O@????_@?A?W???A?C?GB_???C?G?ON????C?G?O^????A?C?G^_????_@?ANw????C?G?R~?????O?_@^{?????_@?B~{?????_????A?????O????C_????C????BC?????_????wO????A????F__????C????^?_????C????~?O????A????~_C?????_???^w?_????C???F~?A?????O???~{?C?????_??A???C?????_??A?G?A?????O??@?K??_????C???OF??C?????_??A@w??O????A???GN_??_????C???O~???_????C???P~???O????A???H~_??C?????_??A~w???_????C???^~???A?????O?_@?????C?????_@?A?G???C?????_@?A?W???A?????O?_@?[????_????C?G?ON????C?????_@?ABw????O????A?C?G^_????_????C?G?P~?????_????C?G?R~?????O????A?C?J~_????C?????_@?B~w',
5811
'K03K04': 'K~`HW}GPHDaN',
5812
'K03K04K04': 'o~`HW}GPHDaNaGPCcPWaN_?@??H??W_?[G?G@?@GC?D_G?N?GAG?C@C_@?PW?GAN??aG_?@CPG?@CPW??aG}??G??O?@??H??C?@a??G?FA??G?G@??C?C_O?@?@WA??G?N?G??_G_?O?@?PG?O?@?PW?G??_G{?A??GaG??O?@CPG?@??CPD_?A??GaN',
5813
'K03K04K05': '{~}AHKVB{GGPGRCJ`B{GOO`C`AW`AwO`}???O??H??@a??FA??N@??G?O?AGA??R?G?@[?O?Bw?O@A??G?`C?A?GR??O@Aw?@?CN_?ACGO??ACGP??@ACH_??O`Aw??ACG^_??G??@???O??H???O??W_??G??[G??A??N@???O?A?C??@??G_G??A??R?G??A??V?C??@??N_@???O@A??G??A?GP??_??G?`K?@???O@Aw?@???O@Bw??_??GO`???G??ACGP??@???O`AW??C??@ACJ_??G??ACG^',
5814
'K03K04K06': '~?@G~~{ACbCwV_~__OOcCW_fAA{CF{CCAAAC__bCCCwOOV___~_???O???c???W_??FA???{C??BwC??C?A??AC?_??b?C??Cw?O??V_?_??~??_?__??O?OO_?C?CCW??_?_f??A?AA{??C?CFw??CCCC???AAAAC???___b???CCCCw???OOOV_???___~_???_???O???O???c???C???W_???_??FA???A???{C???C??BwC???C??C?A???A??AC?_???_??b?C???C??Cw?O???O??V_?_???_??~??_???_?__??O???O?OO_?C???C?CCW??_???_?_f??A???A?AA{??C???C?CFw??C???CCCC???A???AAAAC???_???___b???C???CCCCw???O???OOOV_???_???___~',
5815
'K03K04K07': '~?@S~~~{@@GWb`N@^?~oGA@@GCKOGwOHwGD{A@~_OGA@?_cA@BCA@FA@?f__OJwCA@~_???@????H????W_???[G???N@???BwC???^_G??@??G??@@?C???`_@???Gw?G??@N??_??D{?@???Nw?@??OG???_?GCC??G?A@B??@??OGw??C?@?f_??G?A@^???G?A@~???CA@?_???@?_OGG???GCA@B????_OGC[???@?_OHw???@?_OJw????_OGF}????G????O???@????H????C???@a????G???FA????G???N@????C???N_O???@???FwA????G??@??G????_??CC?O???@???GW?O???@???Gw?G????_??C{?A????G??@^??O???@???Nw?@????C?@?_??A????G?A@@??A????G?A@B??@????C?@?b_??O???@??OHw??A????G?A@^???G????_?GF{???O???@?_OG????O???@?_OGG???G????_OGCK???A????GCA@F????O???@?_OHw???@????CA@?n_???A????GCA@~',
5816
'K03K04K08': '~?@_~~~~}?OH@aFAN@N_VwB~_G@?OH?OW_G[GAN@?RwC@^_GB~_GA?OC@?c@?OW_GAFA?_G{C@?RwC@?VwA?_N}?????O????H????@a????FA????N@????N_O???FwA???@~?G???G??O???OG?O???OW?G???G[?A???AN??O???Rw?@???@^_?A???B~??A??GA???@??C@?_??O?@?OW??A??GAF???G??_G{???O?@?Rw???O?@?Vw???G??_N{???A?_GA?????OC@?OG???@?OC@@_???A?_GAF????A?_GAN????@?OC@N_????OC@?Vw????A?_GB~_????G????@?????O????H?????O????W_????G????[G????A????N@?????O???BwC????@????^_G????A???@~?G????A???A??C????@???@?_@?????O???OW?G????A???AF??_????G???G{?@?????O???Rw?@?????O???Vw??_????G???N{??G????A??GA???@?????O?@?OG??C????@??C@@_??G????A??GAF???G????A??GAN???C????@??C@N_??@?????O?@?Vw???G????A??GB~????_????GA?_G????@?????OC@?OG???@?????OC@?OW????_????GA?_G[????G????A?_GAN????@?????OC@?Rw????C????@?OC@^_????G????A?_GB~',
5817
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5818
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5840
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5847
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5850
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5851
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5852
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5853
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5855
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5856
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5857
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5859
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5862
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5863
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5864
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5865
'K05K11': 'v~~~~~~~~~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~w',
5866
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5867
'K05K13': '~?@@~~~~~~~~~~~~~_?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~{',
5868
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5869
'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',
5870
'K05K16': '~?@O~~~~~~~~~~~~~~~~~~~~_?@??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??~~{',
5871
'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??^~~',
5872
'K05K18': '~?@Y~~~~~~~~~~~~~~~~~~~~~~~~~{??A??C_?BC??wO?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~~{??C??C??A??A??A??C_??_??_?BC??C??C??wO??O??O?F__??_??_?^?_??_??_?~?O??O??O?~_C??C??C?^w?_??_??_F~?A??A??A?~{?C??C??CB~w?C??C??CF~w?A??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??C?BwC??C??C??C?FwA??A??A??A?F{?_??_??_??_B~?C??C??C??C?~w?O??O??O??OF~_?_??_??_??_^~??_??_??_??_~~??O??O??O??O~~_?C??C??C??C^~w??_??_??_??f~~??A??A??A??A~~{??C??C??C??F~~w',
5873
'K05K19': '~?@^~~~~~~~~~~~~~~~~~~~~~~~~~~~~{??@??@G??W_?B`??N@??^?_?^_G?Nw@?B~?C?^{?G@~w?GB~w?CB~{?@@~~??G^~w??b~~_?@N~~??@^~~???~~~o??G??A??@??@G??C??KO??G??wO??G?@wG??C?@{A??@??~?O??G?Nw@???_@~_A??@?F~?A??@?N~?@???_N~_?O??GF~w?A??@@~~??G??CN~{??O??G~~w??O??H~~w??G??D~~{??A??@~~~_??O??G??A??@???_??c??A??@??BC??A??@??FA??@???_?F__??O??G?BwC??A??@??~?O??G??C?F{?_??O??G?^w?_??O??G?~w?O??G??C?~{?C??A??@?^~??_??O??GF~w?A??@???_~~_?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??G?BwC??A??@???_?^_G??C??A??@?@~?G??C??A??@?B~?C??A??@???_B~_@???_??O??G@~w?G??C??A??@?^~??_??O??G??CB~{?@???_??O??GN~w?@???_??O??G^~w??_??O??G??C^~{??G??C??A??@N~~??@???_??O??J~~w??C??A??@???~~~_',
5874
'K05K20': '~?@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??H???O??W_??G??[G??A??N@???O?BwC??@??^_G??A?@~?G??A?B~?C??@?B~_@???O@~w?G??A?^~??_??GB~{?@???ON~w?@???O^~w??_??G^~{??G??AN~~??@???R~~w??C??@^~~_??G??B~~~_??G??A???O??C??@???c??@???O??W_??G??A??FA???_??G??{C??@???O?BwC??@???O?FwA???_??G?F{?_??G??A?B~?C??@???O?~w?O??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??@???O?FwA???_??G??A?@~?G??A???_??G?N{?O??C??@???O?~w?O??C??@???O@~w?G??A???_??G@~{?A???_??G??A?~~??O??C??@???ON~w?@???O??C??@@~~_?A???_??G??AF~~??A???_??G??AN~~??@???O??C??@N~~_??O??C??@???V~~w??A???_??G??B~~~',
5875
'K06': 'E~~w',
5876
'K06K06': 'c~~{ACbCwV_~__OOcCW_fAA{CF{CCAAAC__bCCCwOOV___~____OOOOcCCCW___fAAAA{CCCF{CCCCAAAAAC____bCCCCCwOOOOV_____~',
5877
'K06K07': 'i~~~{@@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@?_ONw',
5878
'K06K08': 'o~~~~}?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~',
5879
'K06K09': 'u~~~~~~_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?~w',
5880
'K06K10': '{~~~~~~~{?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~',
5881
'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',
5882
'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~_?_?_?_?_?~~',
5883
'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',
5884
'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~~',
5885
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5886
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5887
'K07': 'F~~~w',
5888
'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@~',
5889
'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{',
5890
'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',
5891
'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',
5892
'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?_@~{',
5893
'K07K12': '~?@S~~~~~~~~~~~_?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~_?_?_?_?_?_?~~',
5894
'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',
5895
'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?@~~_',
5896
'K08': 'G~~~~{',
5897
'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~',
5898
'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~',
5899
'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?~{',
5900
'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?^~',
5901
'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~_?_?_?_?_?_?_?~~',
5902
'K09': 'H~~~~~~',
5903
'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~',
5904
'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',
5905
'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',
5906
'K10': 'I~~~~~~~w',
5907
'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~',
5908
'K11': 'J~~~~~~~~~_',
5909
'K12': 'K~~~~~~~~~~~',
5910
'K13': 'L~~~~~~~~~~~~~',
5911
'K14': 'M~~~~~~~~~~~~~~~_',
5912
'K15': 'N~~~~~~~~~~~~~~~~~w',
5913
'K16': 'O~~~~~~~~~~~~~~~~~~~~',
5914
'K17': 'P~~~~~~~~~~~~~~~~~~~~~~{',
5915
'K18': 'Q~~~~~~~~~~~~~~~~~~~~~~~~~w',
5916
'K19': 'R~~~~~~~~~~~~~~~~~~~~~~~~~~~~w',
5917
'K20': 'S~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{',
5918
'Lem': 'GqKp_{',
5919
'LemK02': 'Or`H?cI@WS?hA_@O_TOAj',
5920
'LemK03': 'W{S{ACbC?O_b?c?Q_CwC_?QC?cWC_?AOC?cB?Cc_?QQ_?cf',
5921
'LemK04': '_~`HW}?OH@aFA?@?_OWAF?G_?PG?PW?G{?a??COG?P@_?aF?G_??CO?_@C?W?G_F??aG_?@CPG?@CPW??aG{',
5922
'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^',
5923
'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_???___~',
5924
'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{',
5925
'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~',
5926
'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~',
5927
'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?~{',
5928
'LemK11': '~?@W~~~~~~~~~o?O@G@a?wON@@{AFwANw@N{?V~?B~{???G??@G??BC??B`??@wG??^?_?B{@??Nw@??^w?_?^{?G?N~?@?????C???_?G??B??G??F??C??F_?@??Bw??G??~???_?F{??@??^w??@??~w???_?~{???G?O????@?A?G???C?G@_???G?OF????G?ON????C?GN_???@?AFw????G?P~?????_@N{????@?A~w????@?B~w???O?_??????C?G??@????_@???W???A?C??B_???C?G??N????C?G??^????A?C??^_????_@??Nw????C?G?B~?????O?_?^{?????_@?@~w???O?_????????G?O????C???A?C????B????O?_????w???@?A????F_???A?C????^????A?C????~????@?A????~_????O?_???^w????A?C???F~?????G?O???~{?????O?_@?A???????O?_@?A?G?????G?O?_@?K?????A?C?G?OF??????O?_@?A@w?????@?A?C?GN_?????A?C?G?O~??????A?C?G?P~??????@?A?C?H~_??????O?_@?A~w??????A?C?G?^~',
5929
'LemK12': '~?@_~~~~~~~~~~~_?O?c?W_FA?{CBwCFwAF{?b~?C~w?V~_?~~_???O???c???W_??FA???{C??BwC??FwA??F{?_?B~?C??~w?O?F~_?_?^~??_?????O???_?C???W??_??F??A???{??C??Bw??C??Fw??A??F{???_?B~???C??~w???O?F~_???_?^~????_?_?????O?O?_???C?C?W????_?_F????A?A?{????C?CBw????C?CFw????A?AF{?????_?b~?????C?C~w?????O?V~_?????_?~~????_?_???????O?O???_???C?C???W????_?_??F????A?A???{????C?C??Bw????C?C??Fw????A?A??F{?????_?_?B~?????C?C??~w?????O?O?F~_?????_?_?^~????_?_?????????O?O?????_???C?C?????W????_?_????F????A?A?????{????C?C????Bw????C?C????Fw????A?A????F{?????_?_???B~?????C?C????~w?????O?O???F~_?????_?_???^~??????_?_?_?_???????O?O?O?O?_?????C?C?C?C?W??????_?_?_?_F??????A?A?A?A?{??????C?C?C?CBw??????C?C?C?CFw??????A?A?A?AF{???????_?_?_?b~???????C?C?C?C~w???????O?O?O?V~_???????_?_?_?~~',
5930
'LemQ2': '_r`HOm?OH@ABA?@?_OOAB?G_?PG?PO?Gk?a??COG?P@??aB?G_??CO?_@C?O?G_B??aG_?@CPG?@CPO??aGk',
5931
'LemQ3': '~?@?r`HOm?OH@ABAG@C_POAJ_?@??H??O_?KG?G@?@GC?D?G?J?G???C??_@??O?G?B??_?_?@?@G?@?@O??_?k??GA???@?OG??C@@???GAB???GAG???C@C_??@?PO???GAJ??A?_????C@??G??C@??O??A?_?K???_G?G???C@?@G???OC?D????_G?J??A?_?????@?O???_??OC???O??A?_??B???GA???_???OC??@G???OC??@O???GA???k???A?_GA?????OC@?OG???@?OC@@????A?_GAB????A?_GAG????@?OC@C_????OC@?PO????A?_GAJ',
5932
'LemQ4': '~?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???????@?????_??O????O??A????B???G????_???O???@G???O???@O???G????k???A???A?????O???OG???@???@@????A???AB????A???AG????@???@C_????O???PO????A???AJ?????G??_???????O?@??G?????O?@??O?????G??_?K?????A??G?G??????O?@?@G?????@??C?D??????A??G?J??????A??GA???????@??C@?_??????O?@?OO??????A??GAB???????G??_G_???????O?@?PG???????O?@?PO???????G??_Gk?????_?A??????????C??O????G?????O?@????@??????_?A????B??????_?A????G??????O?@????C_?????C??O???@O??????_?A????J??????A??G???G???????C??O???OG??????C??O???OO??????A??G???GK???????_?A???AG???????C??O???PG???????O?@???@D????????_?A???AJ?????G??_????????????C??O???????_????@??C???????O?????G??_??????B??????_?A???????_?????@??C??????@G?????@??C??????@O??????_?A???????k??????G??_?????A???????@??C??????OG??????C??O?????@@???????G??_?????AB???????G??_?????AG???????C??O?????@C_??????@??C??????PO???????G??_?????AJ????????_?A??G??_?????????@??C??O?@??G???????@??C??O?@??O????????_?A??G??_?K????????G??_?A??G?G????????@??C??O?@?@G????????C??O?@??C?D?????????G??_?A??G?J?????????G??_?A??GA??????????C??O?@??C@?_????????@??C??O?@?OO?????????G??_?A??GAB??????????_?A??G??_G_?????????@??C??O?@?PG?????????@??C??O?@?PO??????????_?A??G??_Gk',
5933
'Pet': 'IheA@GUAo',
5934
'PetK02': 'SrGWOMAO[?G@G?C?`@?GH?a_@DG@D??ac',
5935
'PetK03': ']{Sw_SF?_A_F_COAc?{??O?C_?W_??O?CC?B?_C?A?O_C?b?CC_?AAO_?_cW?CC_??OQC??_cW',
5936
'PetK04': 'g~`HW{G@GD_N?G?C_@W?N_?`?@H?@W_?}???O??H??@a??FA???@???_O??WA??F?G?G??O?OG?O?OW?G?G[?A?a???OCOG?@?P@_?A?aF??A?a???@?P?_??OCOW??A?aF',
5937
'PetK05': 'q~}AHKVBwG?P?R?J_Bw?O?@C?AW?Aw?@}??OO?AH??Ha??VA??^_???G???H???BC???[G??@wG????C???C@???B?G???w?_??F_@??G??@??G@??_?C@_?G?@?w?@??GN??C?O_???G?`?G??G?`?W??C?O_[??@?CGN???G?`?????_AC?_??@?CGB???@?CGF????_ACF_',
5938
'PetK06': '{~~{ACbCwV_~?_?O_CW?f?A{?Fw?C??AC??b??Cw??V_??~_??_O??Oc??CW_??fA??A{C??F{????A????C_???BC????wO???F__???^?_?????O????_C????W?_???F?A????{?C???Bw?C??C???A??A?C??_??_B??C??C?w??O??OF_??_??_^???_?__????O?OO?_??C?CC?W???_?__F???A?AA?{???C?CCBw???C?CC?????A?AA?C????_?__B????C?CC?w????O?OOF_????_?__^',
5939
'PetK07': '~?@E~~~{@@GWb`N@^?~_G?@@?CK?Gw?Hw?D{?@~??G???__?@B??@F???f_??Jw??@~_??C@???GH???GW_??C[G??@N@???JwC???~o?????O????@G????@a?????wO????N@????@{A????FwA??????@?????C?O????B?A?????w?G????F_?O????^??O????~??G??@????A???O?G??O??A?B??@???G?[??A???O@w??A???OBw??@???GB{???O?CA?????A??_O?G???G?A@?@_???O?CA?F????O?CA?N????G?A@?N_???A??_OFw????O?CA??????@??OG?C????A??_O?W????A??_O?w????@??OG?{?????O?CA?^?????A??_OFw',
5940
'PetK08': '~?@O~~~~}?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~_?????@??????H??????W_?????[G?????N@?????BwC?????^_G????@~?G???????C??????_@??????W?G?????F??_?????{?@?????Bw?@?????Fw??_????F{??G???G????@???@??G??C???C?@_??G???G?F???G???G?N???C???C?N_??@???@?Fw???G???G@~????_?A?_?????@??C@??G???@??C@??W????_?A?_?[????G??_G?N????@??C@?Bw????C??OC?^_????G??_G@~?????G??_G???????C??OC??_????@??C@??W?????G??_G?F??????_?A?_?{?????@??C@?Bw?????@??C@?Fw??????_?A?_F{',
5941
'PetK09': '~?@Y~~~~~~_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~_??????A???????c??????BC??????FA??????F__?????BwC??????~?O?????F{?_?????^w?_????????O??????C?C??????B??_??????w?A??????F_?C??????^??C??????~??A??????~_??_?????^w??C????_?????O???A??C???_???C??W???_???C??w???O???A??{???C????_?^????_???C?Fw???A????O?~_???C????_B~????C??C?_??????A??A?O??_????_??_C??W????C??C?_?F?????O??OA??{?????_??_C?Bw?????_??_C?Fw?????O??OA?F{?????C??C?_B~??????_??_C????????A??A?O??_?????C??C?_?B??????C??C?_?F??????A??A?O?F_??????_??_C?Bw??????C??C?_?~???????O??OA?F{???????_??_C?^w',
5942
'PetK10': '~?@c~~~~~~~{?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~o???????A???????@G???????KO???????wO??????@wG??????@{A???????~?O??????Nw@??????@~_A??????F~?A?????????@????????_?O???????W?A???????F??G???????{??O??????Bw??O??????Fw??G??????F{??A??????B~???O??????~w??@????@??????A????A??@???A????A??B???@????@??B_???O????O?@w???A????A??^????G????G?B{????O????O?Nw????O????O?^w????G????G?^{????A???_A????????O??C?O??G????@???O@??@_????A???_A??F?????A???_A??N?????@???O@??N_?????O??C?O?Fw?????A???_A?@~??????G??A?G?N{??????O??C?O?~w??????O??C?O?????????G??A?G??C??????A???_A??B???????O??C?O??w??????@???O@??F_??????A???_A??^???????A???_A??~???????@???O@??~_???????O??C?O?^w???????A???_A?F~',
5943
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5944
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5972
'Q2K19': '~?@K~~~~~~~~~~~~~~~~~~~~~~~~~~~~{??@??@G??W_?B`??N@??^?_?^_G?Nw@?B~?C?^{?G@~w?GB~w?CB~{?@@~~??G^~w??b~~_?@N~~??@^~~???~~~o?????A?????@G?????KO?????wO????@wG????@{A?????~?O????Nw@????@~_A????F~?A????N~?@????N~_?O???F~w?A???@~~??G???N~{??O???~~w??O??@~~w??G??@~~{??A???~~~???O??G?????@???_??_??A??@??B???A??@??F???@???_?F_???O??G?Bw???A??@??~????G??C?F{????O??G?^w????O??G?~w????G??C?~{????A??@?^~?????O??GF~w????@???_~~_????A??@B~~?????A??@F~~?????@???f~~_?????O??J~~w?????A??@~~~',
5973
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5974
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5976
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5980
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5982
'Q3K08': '~?@?~~~~}?OH@aFAN@N_VwB~_?@??H??W_?[G?N@?BwC?^_G@~?GA??C@?_@?OW?GAF??_G{?@?Rw?@?Vw??_N}?????O????H????@a????FA????N@????N_O???FwA???@~?G???G??O???OG?O???OW?G???G[?A???AN??O???Rw?@???@^_?A???B~??A??G????@??C??_??O?@??W??A??G?F???G??_?{???O?@?Bw???O?@?Fw???G??_F{???A??GA?????O?@?OG???@??C@@_???A??GAF????A??GAN????@??C@N_????O?@?Vw????A??GB~',
5983
'Q3K09': '~?@G~~~~~~_A?cBCFAF_bwC~?V{?~{??A??C_?BC??wO?F__?^?_?~?O?~_C?^w?_C??A?OC?C?_W?C?_w?A?O{??_C^??C?fw??OA~_??_F~_?????O?????c?????W_????FA?????{C????BwC????FwA????F{?_???B~?C????_??O???A?_?_???CB??_???CF??O???AF_?C????bw??_???C~??A????V{??C????~w??C??C?????A??A??C???_??_?B???C??C??w???O??O?F_???_??_?^????_??_?~????O??O?~_???C??C?^w????_??_C?????A??A?OC????C??C?_W????C??C?_w????A??A?O{?????_??_C^?????C??C?fw?????O??OA~_?????_??_F~',
5984
'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?~{',
5985
'Q3K11': '~?@W~~~~~~~~~o?O@G@a?wON@@{AFwANw@N{?V~?B~{???G??@G??BC??B`??@wG??^?_?B{@??Nw@??^w?_?^{?G?N~?@?A???C?G?_?G?OB??G?OF??C?GF_?@?ABw??G?O~???_@F{??@?A^w??@?A~w???_@~}???????O??????H??????@a??????FA??????N@??????N_O?????FwA?????@~?G?????N{?O?????~w?O????@~w?G????@???A?????O@??O????A?W?@?????GB_?A?????ON??A?????O^??@?????G^_??O????ANw??A?????R~???G????@^{???O????B~w???O??@???????G???_??C???A???G??B????O??@???w???@???C??F_???A???G??^????A???G??~????@???C??~_????O??@??^w????A???G?F~?????G???_?~{?????O??@?A???????O??@?A?G?????G???_@?K?????A???G?OF??????O??@?A@w?????@???C?GN_?????A???G?O~??????A???G?P~??????@???C?H~_??????O??@?A~w??????A???G?^~',
5986
'Q3K12': '~?@_~~~~~~~~~~~_?O?c?W_FA?{CBwCFwAF{?b~?C~w?V~_?~~_???O???c???W_??FA???{C??BwC??FwA??F{?_?B~?C??~w?O?F~_?_?^~??_?_???O?O?_?C?C?W??_?_F??A?A?{??C?CBw??C?CFw??A?AF{???_?b~???C?C~w???O?V~_???_?~~_???????O???????c???????W_??????FA???????{C??????BwC??????FwA??????F{?_?????B~?C??????~w?O?????F~_?_?????^~??_?????_???O?????O?_?C?????C?W??_?????_F??A?????A?{??C?????CBw??C?????CFw??A?????AF{???_?????b~???C?????C~w???O?????V~_???_?????~~????_???_???????O???O???_???C???C???W????_???_??F????A???A???{????C???C??Bw????C???C??Fw????A???A??F{?????_???_?B~?????C???C??~w?????O???O?F~_?????_???_?^~??????_???_?_???????O???O?O?_?????C???C?C?W??????_???_?_F??????A???A?A?{??????C???C?CBw??????C???C?CFw??????A???A?AF{???????_???_?b~???????C???C?C~w???????O???O?V~_???????_???_?~~',
5987
'Q4': 'Or`HOm?OH@ABAG@C_POAJ',
5988
'Q4K03': 'o{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',
5989
'Q4K04': '~?@?~`HW}?OH@aFAG@C_PWAN_?@??H??W_?[G?G@?@GC?D_G?N?GA??C@?_@?OW?GAF??_G_?@?PG?@?PW??_G}?????O????H????@a????FA????G@????C_O???@WA????N?G???G??O???OG?O???OW?G???G[?A???AG??O???PG?@???@D_?A???AN??A??G????@??C??_??O?@??W??A??G?F???G??_?_???O?@?@G???O?@?@W???G??_?{???A??GA?????O?@?OG???@??C@@_???A??GAF????A??GAG????@??C@C_????O?@?PW????A??GAN',
5990
'Q4K05': '~?@O~}AHKVB{?G@GBCB`@wGO?`C@AW@Aw?`}???O??H??@a??FA??N@??G?O?AGA??R?G?@[?O?Bw?O@???G?_C?A?GB??O@?w?@?CF_?A?GO??A?GP??@?CH_??O@Aw??A?G^_?????@??????H??????W_?????[G?????N@?????A?C?????G_G?????R?G?????V?C?????N_@????@???G????G@??_????_K?@????@?w?@????@@w??_????`???G????GP??@????@AW??C????CJ_??G????G^???G??A??????C??@???_??@???O??W???G??A??F????_??G??{???@???O?A????@???O?AG????_??G?@K????G??A??V????@???O?Bw????C??@?C??????G??A?G@?????G??A?GB?????C??@?CB_????@???O@@w?????G??A?GO??????_??G?`C?????@???O@AW?????@???O@Aw??????_??G?`{',
5991
'Q4K06': '~?@_~~{ACbCwV_~_?O?c?W_FA?{CBwCC?AAC?_b?CCw?OV_?_~_???O???c???W_??FA???{C??BwC??C?A??AC?_??b?C??Cw?O??V_?_??~??_?_???O?O?_?C?C?W??_?_F??A?A?{??C?CBw??C?CC???A?AAC???_?_b???C?CCw???O?OV_???_?_~_???????O???????c???????W_??????FA???????{C??????BwC??????C?A??????AC?_??????b?C??????Cw?O??????V_?_??????~??_?????_???O?????O?_?C?????C?W??_?????_F??A?????A?{??C?????CBw??C?????CC???A?????AAC???_?????_b???C?????CCw???O?????OV_???_?????_~????_???_???????O???O???_???C???C???W????_???_??F????A???A???{????C???C??Bw????C???C??C?????A???A??AC?????_???_??b?????C???C??Cw?????O???O??V_?????_???_??~??????_???_?_???????O???O?O?_?????C???C?C?W??????_???_?_F??????A???A?A?{??????C???C?CBw??????C???C?CC???????A???A?AAC???????_???_?_b???????C???C?CCw???????O???O?OV_???????_???_?_~',
5992
'Q5': '_r`HOm?OH@ABAG@C_POAJ_?@??H??O_?KG?G@?@GC?D?G?J?GA??C@?_@?OO?GAB??_G_?@?PG?@?PO??_Gk',
5993
'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',
5994
'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',
5995
'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',
5996
'W05': 'Dl{',
5997
'W06': 'Ehfw',
5998
'W07': 'FhENw',
5999
'W08': 'GhCKN{',
6000
'W09': 'HhCGKF~',
6001
'W10': 'IhCGGE@~w',
6002
'W11': 'JhCGGC@_N~_',
6003
'W12': 'KhCGGC@?K?~~',
6004
'W13': 'LhCGGC@?G?o@~~',
6005
'W14': 'MhCGGC@?G?_@_@~~_',
6006
'W15': 'NhCGGC@?G?_@?@_?~~w',
6007
'W16': 'OhCGGC@?G?_@?@??o?N~~',
6008
'W17': 'PhCGGC@?G?_@?@??_?K?@~~{',
6009
'W18': 'QhCGGC@?G?_@?@??_?G?@_?F~~w',
6010
'W19': 'RhCGGC@?G?_@?@??_?G?@??E??N~~w',
6011
'W20': 'ShCGGC@?G?_@?@??_?G?@??C??K??N~~{'}
6012
6013
for d in class0graphs_dict:
6014
g = Graph(class0graphs_dict[d])
6015
g.name(new = d)
6016
add_to_lists(g, class0graphs, all_graphs)
6017
6018
class0small = [g for g in class0graphs if g.order() < 30]
6019
for g in class0small:
6020
add_to_lists(g, problem_graphs)
6021
# Load the Sloane graphs
6022
def load_sloane_graphs():
6023
dc64_g6string ="~?@?JXxwm?OJ@wESEYMMbX{VDokGxAWvH[RkTAzA_Tv@w??wF]?oE\?OAHoC_@A@g?PGM?AKOQ??ZPQ?@rgt??{mIO?NSD_AD?mC\
6024
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\
6025
??Gx?OyF_?AKOCN`w??PA?P[J??@C?@CU_??AS?AW^G??Ak?AwVZg|?Oy_@?????d??iDu???C_?D?j_???M??[Bl_???W??oEV?\
6026
???O??_CJNacABK?G?OAwP??b???GNPyGPCG@???"
6027
dc64 = Graph(dc64_g6string)
6028
dc64.name(new="dc64")
6029
add_to_lists(dc64, sloane_graphs, all_graphs)
6030
print "loaded graph dc64"
6031
6032
from cStringIO import StringIO
6033
from base64 import b64decode
6034
from gzip import GzipFile
6035
6036
dc1024_gzip = """
6037
H4sICD6oslkAA2RjMTAyNC50eHQArX2Lehs3zuijcSiONZZdZeT4oqRyS8nOWnETr+Lf2bXdbs+z
6038
HwLgBbyNRnbxpep4hgTBGwiCIPD/RC+E6Jbf5u3z3eZEmj8a3a/atj+db3bz9fN099didSK77+J4
6039
/Ze+kn92kONC/ila8YcWz9NfxUR8Wgkl1p+EFKoxGJQS8D9IqTt42d6aP/tfBWR9gAIJ+l938L+j
6040
a/xrcYq5NGJobMbdqRZCTs4nmHe+hgxKvEw15vjPB0JmUS7mJ/h/1b/Siy/TH/h/+e3qlL5svy/p
6041
QX/FskWz+/wXJV62H+hBisXK09j5X0+3FK5QqV1NxGf/6TdHkxaEGSrUPtuUpmmWHvnl2j9OXjxW
6042
ceWQKdF5SnrR/senbtpAoZj8W4c/Hp/Dszrxj/KpZxm+6i58+VMHgprr7jSkWzzOXdm6aTYhj/l0
6043
/4H9tfnrmv21Xn1kf/U3W/aXWv6Po7lrf2V/Necfv7E/pf6yY39O9PUzT6xP/3IJzb+uZxWEgsQm
6044
ZL7ciOjrtRlf2C0WrtREi0kg7GTd9KYA4d7I+eLcjBREi7C4n/eCWkAjMerZjDSkh9DK9oe0PW1f
6045
iGvzT/kieslIoic93erpo6sRFvaHEg/UfHYcylUvXzTL1KsH3V4KlqafPOjpB9cK+L75spvQlPHU
6046
LD6L9RLKaVw2JA+etbKVEsvmaQtzWcrO5mt28nX32ROuJphDIlp15FMZIvslvny1dK7PxeZpRdXs
6047
pG1VreY32Ebqwo7I7lH0f59g8d2dpWwhzZA6sTV5su8Mxmln6/hK1TzW4vhmZUk/oakln82/u6XN
6048
/JlynGtkYNToak45TIf3sxffRo2pKT7a2W6eLFc50mIq/rSDS3dXVI4GNvhk+6Fpd66LtAqTwUxC
6049
6dmHh+kC6kkkav/2BIYsDrl4YMJbbNFJ4EhqBdwSx8lUCfe+Odo20P7w+EIN5Zv1CJN88DPEMTJg
6050
Wn5iGEL1rRsvxNLdwDwxY0QiWa2mQUQfzRhfYDGv1HRuYHda7aCID2ssY0NYcHhD8mb1Cf6WbJBv
6051
IWv7Bedg74b6XLfYRDPkS3rnRu5cT/H96b/xxc4N+3Y3QRLUnaYXlkwz+DWmWdgZahEBZwBEr47y
6052
Hqb2uSb2rwPH2LkJKqZn8HW+1pihCR/an/Dh5EZ/xTdbV7YwU0NC1Xpa7+5dIxk2+mR+m+bp0jaa
6053
dFlMxQ3GzizHiHyp/arU/42D7G+a3d061GS9ApQLN3E0+2K6w3xpibGorRsYZgbdnJvvsNBjZS5F
6054
IPr5TsEvW5rEws5t4BqdG4zNjGXaXin8pcEtexFaR8CygCIEtVv4NBU3t/D7p12WdgHhRCwuBckb
6055
vgFcrcxE0/Br1zzfV5KQ9n3gvX6J6wC7uG7E1k89v8T0ZsEx0oqUOz8Bt+GbqYv4tjEz0U7D5jzg
6056
bIGsa9OuU/fGjJ2bWzfHNAykUFu/CmjxGXvf8KPf7ItPoRLNmQBx6FE0T7Z6XnYwUxOa+WRt5qQb
6057
z+zjGj6K70pYochORUq42MDvh4347ZZobVae2/TqWcPsetZ2lW7EbBEoukC8V1JY9qrE1jdx18Cc
6058
7ZqrTl5ZOpptQDyB2dJPvveTf7uvPfsq8OvxbuJlDEPHjVJOasHFxi/OpraepSqc1woGv1v+O7bO
6059
06TXfsWAP4K8I2EwGHS7ya3lIOaPIHKAjACyqGkNJzE1YUiYNRI/y169eoFq7UeFkdhwmLSdnPYO
6060
Z8ekpgUu/It7sXhytW4Y45enkF3Oevlh7bJLmMYOFXZ4Z5aS7617q5jMQ/3Wa7GYBJIbeNF71pYu
6061
TIpJX2eIdKHFRyeoAtcO31taxp9Fu3vwBcwCfeIL/l4I8bX9P/+SSYaS+KeZWfrCy6MNowBbH36P
6062
9IvvlIaVMHG/rfgReo0laNyvqfeDf9vxj0w2DeK+R6GjNqK08OIyfWlRhupp0z6KsLLpH/pH86bg
6063
kumR52+Tc8HBEzJ3a0EPjKyUZrL0f510PW8RX9ZVIKTbsLllAYgIknm3sDlxFXBppXgJjaN+uAdc
6064
G3w3/L/QaPLZExItr34LYBLdB54tVEjkZDcgoo3q4yjQzcXOv+jEf4XWaUmGstCzovkRV9khuvs9
6065
vDN7UhUlslU/YmmQxRRaTx77shrLkpJUYsqmsxOs/YbDwc9T1sf9ylU+6nh5vWObOCfCJcjkK9v2
6066
ic0txxCGPm24LSzu2YgVvk+bafdXeGmkpYRkSmb26uyFGcwyYeY9Su28ANWlqKhb3DafwIzotCWx
6067
Bpspx9XtCqlMgU5DYEt8Ttub4Gpj91wE96IppVLbf7MmNWyoVKQQt+2c/7kSyYy0pHm9BMGuXOZE
6068
fOWbYkBVwiatpIvQu39JiY9dVISRJjQTmuwshjn/OcoNO58CcUdbWFpDOolVzZv34waEv1tWLPDC
6069
vOWuFO6xwqYHdzx5usWTWXZIgrblNo+lcrsXM0BIbHYC+FIX6iFnP25F82yrQAXncxwKnpuhaKVt
6070
/O1l1xe6ozG7BXFJr23JaqsLCaX43fzreMJ2WU5opLCr34Sj8dK9VkEuF/Fq6uB41zMuGHjjB9E7
6071
duSWzbNNLiLQprq5pqTUyKrpulL76NZs36zI4DZou9JAFPqXazH/vzilLqecXYnTO1ddSl/gwFAz
6072
afasLe6CvHqjE6Uamc8nC8uWLAPqULYoFH8qXpeN3dLaLlqJ0mwAxcmfmpSZvvxtMakZ5+1TT7qZ
6073
J5tUVpLOtFoteLc2Wk94CidqXAp97PdRK7d/fC1Wq5lsm6NzGk6bb7b7URApdP8n0d2Rhk8eu81v
6074
D0QXJoip+AOtg14Cp0GWt+zG7IQsA+f6nWLafxmKp19t3a6IBlLAFLiI2UhsviMiMyC3+ABbq1Ld
6075
jMQut35zsyblzkyUiTBNcK6sVAkyPj5QFp2mNUJss5tb9t4rUneRVqfAR3HDmzB5Lvd62iUM16Cw
6076
wiEfCCiMnmRdOEFFFOc3BC8wYLPsr7BaTFjxltwtzBmSVdn25RnerpK3wPxhL419xarU7mCAbjNC
6077
roAQYldskDPtV5R6AZOH9i1swZW/9NLuoSNKuheTmbptGigRm5XfhvNaypnZITa4aDWRwt3rxaIV
6078
ZzE3yGmoc9Gln2x7t4ljlOMC0aOObL4JBNK/bJ+icTzk/WOwLMTx2j4zGmE6/QsR/M2bC9aW1moG
6079
NKspri1YmSu+A1HNvGvOw+lIQLPVLSrs2D4P3pvN7szvwBg1Zl5tsWnvOR+T0H3PdkFh72FqTZB9
6080
Te4YlaDuf250Tg7MriNs3CN+YAHpn5zaL0oPywTpDHlPofJpqwv04OFCF/UU1dfMkPPGP3M8XzX+
6081
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uW5MFKjnPaufaLioh0Grlu9XrbsSRtqzEnSozywu8X2YS0aSf8CQ9E9COVXxe4FErIO4aElZ0Ikg
6343
rPVoJUJK//cbijhIbKX3QsVyqAmyn63MBB1niOXBzv9roA6znG0q6l/FdNyEcr4GuXi9e7flq4Pu
6344
MIWpjD1oMDTBgwahvJJwWNX2Y9wnjAKD9hCpoWGufmM0QT3e4fnAlZS9mEy2X95NogUl5EFyg7ae
6345
izI0zTlDaajuvredmMrX8XrpPWAa6VEfkL6rqMR9pC5vTT1d3Ivm+W43Prj33qLV7QHJGxuXKwWz
6346
b2cbwSMUS40M8OH/Awl0aLwEVQEA
6347
"""
6348
f = GzipFile(mode='r', fileobj=StringIO(b64decode(dc1024_gzip)))
6349
dc1024_g6string = f.read()
6350
f.close()
6351
dc1024 = Graph(dc1024_g6string)
6352
dc1024.name(new="dc1024")
6353
add_to_lists(dc1024, sloane_graphs, all_graphs)
6354
print "loaded graph dc1024"
6355
6356
dc2048_gzip = """
6357
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6359
v5Oz+7/4DfuzJR0h1+xPMiVfOHmZ/4NMyPsNacj9pn+ZCAr9Hz0m4oN8EB/zhojXOvH8T/h6ySHx
6360
LfzayR8e4Zcv8B35JP+5fIF/Jv8P/iHvfwD1Vv5Fmnfwz2sj/9z8CqUh039wRWQtmWDzTtJhfYnk
6361
w8/qgU53X+Xbq8mOSBYNriY/1NPT9A/19Nv0X+ppzb6opxn/8aIed7Of1RMlHVCe7Dqmvpo8djMi
6362
uafv2jkhF5zKGuszhq+bF74mZA9fdq/Tjizlj6vnvsb+IemuNhec3EiK7f9rOjLlskzkbkY0w93k
6363
pZv/0Jyc35MrVUxGbvr/bvUf1x/ItchOVmrPjq4BPl/y+T91RZNvG/KvnfqrZdM9O+OqGfoi/VBv
6364
i78W5P8uqGC5p0Xac3L1tHok8m9GO/baN42qjtVnuvo3lE+8SCe7s/ntXyo7wmYd+/mbbu5OFGJB
6365
NW8dazp29m1jGuobIdu5bTt22za7/7bm7/O+vrY/uPn7mpCP0z/Nn6Sv8sfp0v7dPy4nNzb9RPz3
6366
24tN0AkezuGx73y8Z07XtOz8fWknff+QLz+K9r/Q70HTUyIbVuTc42rb/8I6lRWBDssIvYTHvayS
6367
u74aZtACc8lY+9S/tICXVJfrB2lHGvhG8tbz8tpSJiqCPWsOl/dktRWPv6gKaummZRvxGvsTXl58
6368
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6369
P6B6P9i27L/740KQe73VtS54uxK1QW42VL/b13nf2fo8ps/dLXp59SxeftguVXXIRlrT/ZlpReCQ
6370
NBve022/P+zxl3fw5acHPRYgn7Oe857tbxfmfQ4dsuec4m7ZJ9qJyaq9/T9boP7pevvSCZ5+5+bb
6371
TnTRXf/3dYM6av/tYyPmznXzRDCFJRP5LOl/LAUmOpOYlPv5+cW+T2DEcjmocOK+7TbweWu/buC/
6372
8xVpCEYD/Xm+Ep+tQ6SvVfbCl2Jmw1yz/q8n1n9O0BTb9/C1qObVmtA9ISaPloiyrD7OW93VuSIj
6373
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6374
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6375
pepz+eCVRSwz8Hm3oN4vS/V5826i52w5urnpva33TiOaT3xOv/O9+9P9Wn1unzfuL9AhSd/92Xz1
6376
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6377
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6378
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6379
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6380
bW9+vlMVskE//6E7NW96yQOq+oJY4ms98BsqOieUC02jzExi3DRzZ6j30qTtDiA93q9bzQ1zJrb+
6381
6w9bqFdb0eT1gqHfr+TvG/v708Z0CrF2foPu0HLD3rvPZoWlcnj37CxaQ7Tptw0og/YHrLkfbfnu
6382
n+301lPafhVkmkdTP6zfcejf4acXUQPMzkkMzQzUND+zq7fYrBiI8QQL8YSYLOjFz48mQaO7yGpv
6383
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6384
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6385
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6386
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6387
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6388
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6389
rV+ST2sz6TncT/tNoNrcN2ve3lFPHJBZtq/9Vl739nW/iLrynR6qD1ABkvCi3/zyv2LJepkWNkqQ
6390
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6391
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6392
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6393
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6395
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6396
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6397
PutoWrnf0CuymIXF26/zWFohbU92ZrvVTzvwuXqJpRVyxHvLxOov+Py3P+30+2KtESSM6A2kUrRG
6398
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6399
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6400
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6401
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6402
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6403
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6404
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6405
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6406
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6407
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6409
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6410
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6411
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6412
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6413
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6414
s/oUn1jYy4NlFeaV373WWPB4a3xmrgEEb5Yer108y6mSgPGP7/AQjL0k0DqMwLmLOmw032/i/Z3u
6415
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6416
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6417
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6418
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6419
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6420
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6421
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6422
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6423
pFEOXSZe/9p3ixspbaJqVSbjBL8UkVdabXjkwrY6zEZE6wTCquun/s+h3o3rT97INVtKsZGMuLSS
6424
d2E7PFnJXdMcsYWxFFZkv/vvI/a3UqZStlvB+/0EsZ/dBFzp/Pv3N+jNyJCBEyH/W1zRk13XII4s
6425
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6426
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6427
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6428
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6429
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6430
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6431
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6432
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6433
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6434
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6435
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6436
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6437
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6438
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6439
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6440
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6441
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6442
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6443
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6447
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6448
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6449
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6450
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6451
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6452
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6453
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6454
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6455
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6456
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6457
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6458
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6459
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6460
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6461
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6462
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6463
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6464
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6465
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6466
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6467
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6468
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6469
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6470
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6471
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6472
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6473
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6474
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6475
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6476
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6477
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7213
"""
7214
f = GzipFile(mode='r', fileobj=StringIO(b64decode(dc2048_gzip)))
7215
dc2048_g6string = f.read()
7216
f.close()
7217
dc2048 = Graph(dc2048_g6string)
7218
dc2048.name(new="dc2048")
7219
add_to_lists(dc2048, sloane_graphs, all_graphs)
7220
print "loaded graph dc2048"
7221
# Add all DIMACS graphs from the DIMACS subdirectory
7222
def load_dimacs_graphs():
7223
dimacs_g6_base64_dict = {'MANN_a81' : '''
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''',
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''',
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'p_hat300-2' : '''
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10280
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10281
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10282
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10283
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10284
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10285
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10286
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10287
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10288
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10289
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10290
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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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10303
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10304
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10305
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10306
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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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10332
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10340
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10361
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10688
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10689
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10693
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10697
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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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10914
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10915
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10916
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10917
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10918
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10919
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10920
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10921
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10922
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10923
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10924
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10925
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10926
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10927
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10928
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10929
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10930
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10931
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10932
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10933
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10934
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10935
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10936
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10937
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10938
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10939
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10940
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10941
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10942
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10943
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10944
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10945
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10946
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10947
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10948
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10949
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10950
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10951
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10952
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10953
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10954
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10955
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10956
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10957
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10958
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10959
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10960
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10961
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10962
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10963
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10964
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10965
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10966
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10967
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10968
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10969
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10970
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10971
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10972
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10973
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10974
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10975
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10976
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10977
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10978
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10979
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10980
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10981
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10982
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10983
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10984
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10985
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10986
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10987
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10988
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10989
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10990
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10991
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10992
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10993
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10994
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10995
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10996
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10997
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10998
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10999
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11000
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11001
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11002
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11003
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11004
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11005
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11006
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11007
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11008
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11009
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11010
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11011
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11012
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11013
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11014
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11015
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11016
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11017
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11018
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11058
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11059
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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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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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11081
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11083
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11085
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11086
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11095
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11113
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11118
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11430
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11431
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11433
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11434
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11435
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11436
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11437
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11438
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11439
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11440
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11441
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11442
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11443
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11444
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11445
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11446
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11447
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11448
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11449
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11450
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11451
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11452
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11453
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11454
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11455
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11456
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11457
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11458
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11459
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11460
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11461
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11462
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11463
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11464
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11465
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11466
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11467
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11468
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11469
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11470
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11471
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11472
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11475
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11476
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11477
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11485
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11510
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11518
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11519
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11520
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11521
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11523
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11524
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11527
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11529
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11530
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11532
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11533
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''',
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'keller4' : '''
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''',
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'brock400_2' : '''
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''',
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'p_hat700-1' : '''
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13313
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13314
''',
13315
'gen200_p0.9_44' : '''
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13343
bfkMAAA=
13344
''',
13345
'gen400_p0.9_55' : '''
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''',
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'brock800_2' : '''
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13801
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''',
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''',
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'gen400_p0.9_65' : '''
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''',
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14785
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14789
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14791
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14793
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14795
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14796
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14798
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14799
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14800
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14801
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14802
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14803
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14804
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14805
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14806
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14807
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14808
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14809
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14810
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14811
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14812
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14813
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14814
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14815
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14816
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14817
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14818
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14819
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14820
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14821
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14822
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14823
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14824
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14825
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14826
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14827
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14828
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14829
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14830
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14831
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14832
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14833
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14834
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14835
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14836
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14837
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14838
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14839
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14840
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14841
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14842
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14843
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14844
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14845
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14846
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14847
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14848
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14849
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14850
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14851
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14852
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14853
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14854
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14855
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14856
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14857
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14858
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14859
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14860
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14861
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14862
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14863
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14864
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14865
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14866
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14867
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14868
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14869
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14870
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14871
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14872
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14873
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14875
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14876
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14877
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14878
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14879
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14880
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14881
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14882
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14883
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14884
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14885
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14886
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14887
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14888
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14889
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14890
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14891
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14892
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14893
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14894
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14895
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14896
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14897
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14898
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14899
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14900
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14901
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14902
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14903
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14904
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14905
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14906
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14907
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14908
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14909
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14910
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15989
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16010
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16889
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16890
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16893
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16894
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16895
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16901
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16904
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16905
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16906
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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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16918
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16920
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17900
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17901
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17905
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17906
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17907
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17908
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17909
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17910
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17911
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17912
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17913
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17914
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17915
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17916
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17917
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17918
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17919
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17920
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17921
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17922
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17923
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17924
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17925
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17926
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17927
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17928
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17929
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17930
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17931
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17932
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17933
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17934
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17935
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17936
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17937
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17938
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17939
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17940
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17941
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17942
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17943
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17944
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17945
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17946
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17947
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17948
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17949
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17950
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17951
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17952
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17953
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17954
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17955
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17956
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17957
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17958
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17960
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17961
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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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17974
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17975
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17976
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17978
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17979
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17980
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17982
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17983
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17984
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17986
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17989
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17992
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17993
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17997
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17999
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18000
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18001
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18002
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18003
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18004
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18005
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18006
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18007
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18008
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18009
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18010
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18692
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18699
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18701
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18702
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18703
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18704
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18705
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18709
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18710
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18711
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18712
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18713
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18714
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18715
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18716
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18717
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18718
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18719
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18720
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18721
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18722
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18723
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18724
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18725
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18726
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18727
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18728
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18729
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18730
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18731
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18735
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18822
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19474
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19476
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19479
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19488
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19489
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19490
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19491
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19493
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19496
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19551
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19552
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19554
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19561
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19562
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19563
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19564
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19565
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19566
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19567
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19568
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19569
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19570
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19571
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19572
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19573
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19574
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19575
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19576
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19577
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19578
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19579
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19580
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19581
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19582
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19583
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19584
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19585
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19586
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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
1/3auvDikltBiAx+rvvw36neQ01lhHOErYD4pbodFU7yToIYZhxQshhTlTllT1VfgKvfNDjVhNLO
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
OkO7Sxv0Ga6hDZKo1vtFvLxNQY+Kwddg+mPfH5PLc18z5bxx1PMpLKIu6E5S16k1MsMSOvEe0C7i
19721
p/kVZLoaco8gAFHXtcOs5Ehnn66q+5Ng3twgR52u1ziAaY/28VCWOee35DOyEMSvBGhqQsdkbZ+8
19722
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19723
WmZKlGKZyK1WSdsYOmdDPVKEWLvMcknViFbkLUMSBJY7MdLAPU+IdrnsHCOMtzvY8Lp+/lRfSn1F
19724
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19725
fWWqTRi6YqEWQLwhM3rrNn/ABw1PPWeoTzKLtwIQmeQvAb1+f2/WqUzGx5rky7rkEKQeMJeYbu7X
19726
HOQ+7EfdzkLuwxXKGMoYT/GhnzyxQzUUpDJEDzPz6r6fzSAWzg29AkeFHUXoPi1DoXy+UXXWkNSR
19727
SunhBMfgffEY28W8Rsecxw4qh3uw9v9JvD3VlgLxE2/Kukp6LyZyYLGav3QP7wVM3xQoGwt0Ees+
19728
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19729
Q0CdTuRZYt2G04qf4W5hCeXjUahROUl/VvzrGF9HGAO9PNY5VrbtNCQw3YUf6rYlKAlrBiL62xXf
19730
737mZlEnqHQwU1fG6XCcmSoE8Dvkwm0cD3QHJ00ED506cKZYQlPtvWO7eIMyu8SL4kYV1c/mD3k5
19731
nHNJostxt6ppZ12ta/0XgsUVP9ivHeFWzkGiAew5vUvw/1PomtiMlc9o97vl47TS+5DtcaY4pkrh
19732
xc3YWdKDfkGNoBEI7SqJXkcrxxNWS/rVZmdFENCPQzPoREZJl+EpaLtSatU7bEiehGsRpSuxomaC
19733
AjqGk9RfE+5T91NTel1saVh/DYG8NHF9yjVtut7RVisTiWxnadUnVFRvEskFRV60ilRdeMYEZ1Xa
19734
VMxEiOTQ73S++6vXVhwE2Rr8bdAxnSQBwKcySz1cbuzFaY86fnJFPIvaK8Xh208G2tUgXGawJaSm
19735
gEQTX3tctlFmCSaezhzfQOPHGjqTTQYvtOLYEzKHbfZPKn471wWM30otyZeRKzAfrKJsqVBL7pQG
19736
Rcrh/MYw02LRzcPWZOXqGhDufToKMsPgaghcCMyemfEF0m/Wtaf0JQuh5I3bTr7J+ouVlaNJ/BDT
19737
Hpyhj3FaAdXwSJ8yStzKDubS+hf25UGQa0xdq8FoBC3voL627aDHqjfOkVK9oVLAWHySvuZ+eDs1
19738
/HwA5ME6tSjYijo0KLVtzB18XtShiNv06LxuKWT8O5Y3mCmZT2b+B4DQ5TO6WnKsQtUT0eb6AKsQ
19739
icHcfPLtErn7/SSCh+RGxqvMSPOCOW/ic1l+DQxn8fX5CM85U/vwMEz4TD998AvBfvWKojvh09mq
19740
OAyvYE14PZuOfxc2J6DkRmkTHGaVTz/6q3Pq9Tmcq4WSyo1y6IkVEvsE+fLOmGB23x6CcRSksU85
19741
iMJPs+lw8C26aZAL4z/sYjA5vN8plB/U91sNtZo7+QUCVfThW6tm0AWEBCGe4IrGxWrqlE+epS0v
19742
AONm0xN9i4nR0L5EQumMCs9vm/Xwg8GCYeNiZ1lozpJS500MsH/kJPbEk0XSXSxsRi0UXUvR/P+x
19743
QUWf8ri5Dq2NnAsejpn24va8R8ObgwIgTJw+KlpoWKRwuYrz80wkRwKLN6s5Z2NdQWvMVIEbVXLx
19744
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19745
23G19Uzwk9V1MG1Cl2KoZFsTr1IaP+e2KI6d2GxGzvLTI8zVIPHXIG/qUxoSqRUlsqKPbEtBoSkK
19746
BL2xD6fQ9RtUAbhwrh1o5ya/+hHZb0trHzcvRSQt+8kxBYlEAvCgRi8bwM0q1CcsbIq2SiniD+Me
19747
kbMHEDgsjPRKf1drKyXX203ax7hZTUiPYilhSE8KsPj1caIegYRvN8gXrjq4ZKCqE0lY1EzCR5gJ
19748
GAmpZyT4JmFI3skhkhCVkEXzHIhstIyR8aYat0SBEyhr8Ra/Dzo9v0LHzs8cHJxcVlBd0re1VXMr
19749
SgBMQL7YeR/o3VxLBNGMSyYzyVaU/ZiksNOTce6EbG9pwGVEpDLXlSuTDrJQ5dPqAGESoZtiU+Oc
19750
R+OFxXE7THOtyCkjLPZ9CQd1jHONVMQ4GXOnQi2EtHWRZYTLxPTqBC3OV9TaDN6+KGid60rIeHIV
19751
yd2Irk85UgqkZpZeD/pRintaVJGEYLlsj23E3bYLO9D6PuY357/G7qoiIQ9sVZOQUB6mj7JR7sM5
19752
a4cGHCbeb7rhVaI+gEehxjeMO5rjepyoXNk8nEa8TrbIYnarLQLY9h0fZR4vzFgMG5KGJNIdQOII
19753
iM/LXFTF+87mlMrFmPhBhgkrv/1wmauTPv3HeP05ArXz1lw0UNct2sRPOR/xIMqnyTwCvlsJ4XNf
19754
0V7L32peGoV00jmziIrlYjsisrslkRU0gPWN9jdxe0Qra+eBkFtKqqBhc4QTf4x1it9IOLsVCI3a
19755
M/hktBKyyV4JC+RuyfaMLtb4Xp1Nurdvw6wzXj5IdMVl4Rry703nyOobF6Eye+7ZYgf1iixQcx66
19756
6v22fLG33G2I81mDkANBwNXIPAiHI3HYnXjE/LXTkfwCC0RyQZEfCf8jic35tfMkRhVBXVZXAQhJ
19757
zyrti2mO8SIlVGCDbFvy4LDD8oPavh3hb7+WRWZWCN7BGTpd3U/YNvFZIm6Ve7oR3GdH2LhRketR
19758
5hkkOJgkuwxSzV3rHUjCkTsbdlZvCvlWRLbUzj4n9rtRFk5+8Z0IeFXT3n4vDBkkwt6Ivcsk0J1A
19759
BmkxhxJbBuWrpr4T+UcFv0+i6IWWWyA8b2QZGcS5Z+HOGsvVWdzHwjc4xxdS+jBFPc/tadj1jfbf
19760
0oNn5IeWizF/xgYGQKpNGyNEbupWa+xD3y7HWvChwt0eCgmo+W63w8mafrX99OUNFVP+xxlbTE8A
19761
e2N0gKrn9OGD5XelupfcpywqvxGN62r49DoqnT9uSYnaSFYcaPr8wuNXXK18cyJHDovYIbshybye
19762
Es/QvAMd+o72dH6U1vttqPIwm12H6aT8P5jdKU9z+SY02K+IMTOWIB57qkIYu1S8mjMZyGJEoAhi
19763
4r8jowpuq3OZPPOZY68I0wh7/icV1yiKe8M0kUb2d51fz5kqdf2wP3aPdtajyMyKABlj1LuzEV/Z
19764
HpVHUpv++oSwNYAuqbQMlbp4FHRBWTh2XWcsuh24/PB6j05xPDg1CUFyWc8Ozi2Ec5WydFyw3T+/
19765
tJfpW2YCzIF1FPLVkgt+qSrpRy/JmkTPl5l68Wf+0DOfivL0xN1PlW/nYmGV5GH0B2+COnLI/SHQ
19766
jd/AUH6YVlvqCv+AMCgLFBmiwQdR6UQ6qnq1jnIU4u0Aixb8vwMjjL0oSKboRUYfAod7B/w42Jn9
19767
4uCcGhePOZQ66mSDlDRKv+Vw/BJHau6vkmbb1L/FLBNYPfgyQJDxY6+x+EO/VZFMFg0I5Mm+2iWM
19768
9tHbJGrvPxhk2gLmIH3gb9SnPhm3cy/lZwE6LvkSNnSNoDZdB59P33I5ANycyI/0rWgGCtfKgvvr
19769
2HqfJzdKyM+UEJbWGWFXjO6BZeCdtWekb7dtFf0oBo6ZPEyMvqdhpd+Jdj2B+w1yP5kW2u+tbWRn
19770
KC4WESRVHp9Pwn4jzpNQnPmkbC7ttIO06JkXYzfzdz0SYRZ3XvOja4FuUxaZ1rDsvHWKsg9v5lsY
19771
bzR1b2vNTPDYqKTOMIad1UH+c/RZcuC8QBDcp5H4cdTzk7sqwCM8Olchrr/zzHnMVdmEnUQGNcft
19772
DQG24NSwyFDk8bMZMAyv0OwsSYYIFJQtF9Qiv7Vn26U7L8cR0GcCoxu/gH3gY+JIsJBkd0gFvM9p
19773
Of/nouoeAYb9ZY7SGGlW8F2+E/ECxLeTg+GeER1I3rAn9Q8y4p/PNLtQ8WM3Saljfj03HHMlU6FJ
19774
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19775
xARQgw2kdJJWlwCx11nciYXww1abX+Uqimo/2/gZ6Asl7RiIHi5DVdoXr1eC/rI92uz9vvnsAeyr
19776
BgnEWDc9XtaudKLQbI3iAXHhV3JfNxWoS+d33C6fxb2KTNMTrI35+eaPebH27qe/+YYyTXGw4Dan
19777
Dzzt47jm0op+SfJrKCS0yAxboG9Fro5DZOZe7V2pg4e9tyxXzShpffIPlfdfpJhHcAqjajofMjQD
19778
SdzcR4D1z6uc6v05PLh+XEQfgBpdB08DvhzyEd7pN5QecxFrxdAOMPbqDiHNiRnm7+h8XgepacO7
19779
2s8GVslt9ecKBhnosc+A8EXMwZxsTUTJLQ4m+7cmdzYIu5d/BG1mre3h2jnUhdgCp66txuPBNI0B
19780
bX/SoVk6j0drXlq1CTd02r6mLkQb1nch+VzAUf9LGaC0iSRvjfdnmMBiZa0W5g4OVYdTxbMdkr7t
19781
it5xljtAiH8/7fx9YGYLL8FJEETApMaK+sMA7oli/LvWryg4v3mDpUghYSeRoIIewUu3WFQsKtsv
19782
QGS1hh6HoL+qmsU8EEO/35CUubL85KjgF/kTXRtsVeIUGY9rSGWE/o7ZT1zMiyz6yAbgs0vYVN6P
19783
TMjgEOdtyQ7UxWjsM+9W7RBB1mZXEUfdStsIXSbFWDdFBPNIDOhkx4W3pjs3x7xmZRTZbfKkKA4U
19784
mC1K2zk7nxjSPc0nETJzz7OZEEfkjoqIOKhl69u0lMCC630Yp5KMKzjb2VZwq4Robd2yj1DEcV7V
19785
/arEg77sU9qKZeXICZUumk0AR9ZUo2Nr1QAJbamlMDA8zrrd6YeNd7s6tDrBNS3AIxOtUvj5eHIb
19786
ZTi8Wp9GGeaUnd839MhpWixEH/FxBz9+GsaRW0RVCJmnoT/luDNt0+YrgnLOyH8acb8M97gqy7u3
19787
skqUdXSvp7njscK752ZGSpQxQFxoNUbPIGH8fmkaS0XfZSp4qZfUoqZWLwq2ctmBXvBbd5G6YNI6
19788
Mq1l5NJfyEoBDRF0ezAkgLKdhwPnjDKukzxNtahMMfdXZFtonjZiU8gSzyOVzic6+d6YELEzOITq
19789
T4gjKG8BU6Ej8WSVGKddkeJlHgTQggz/8oycVbYc9bB0aCXFBcNSwDNHk7EPtnOLbHNkQTZiwDWl
19790
yw4aMc2VRyFc+PmpfVbSjgo2v7/QoXMMwCpnLgghJZjxNZP9jRWO397mwnXEB+xkrAo38WvfZ4PV
19791
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19792
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19793
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19794
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19795
MmugTSGfB9YIVl/eV0j2ARWfPhrJRuQPhy6rBCm04T2rXY/KtmWQcWtVGEQwtv0hInNzoqURPAyc
19796
FFZc0qWAPm5pYw02sXejWsQt2tHVYgaScxZvy/GPzcqZbW+Ys7SMmvUV193gCmpUX2i3UcUKJr/6
19797
hes2UzikVxO/eDjIXcVvICbeQlOiLP+i9XH6gJz/zrg7KR2uImU4i59oEZ6bTYKpRgEs13xJnkM4
19798
1keT5hoSWXput287TgKGdq4PBdbxKbt9eLZH/CwCNaRUdhFOezuvaoe/yIolCQ9Z6AcYHs4Hfi89
19799
s+2/OdCHoi59Ji1N9c9+uXZ25m1Xifbr0NPouJ74O+rdwIompTv1A/bGAJ01AMbYOOfosraRMSiP
19800
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19801
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19802
JCTC3a4Vl/ySWf+qjkOaCTJYxai793hKGkPm0t1LI3YttSzeFa4FrNotwEuu4OEv487VVZlqyjD0
19803
BCNpBVZRMe3CzbL5Kd+A2Db/oCN+boSMHMEly5DJK8T90GYJrCs0FF8e7b7mfCgmL35LWCuqA+PC
19804
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19805
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19806
xhVFxddLuQPBFhG/Y0hPdg+YMPUVenqoVfCXvCjJAkur+ij2Ywz5tfEfQaO3G0VAb1feIY33uFRH
19807
BF8Dy12YXMBIRXRf2wkPMz/L/OyIwMd+D7Hp3THgyJVL30X/wWkMc7jpdRZoz+zQW0II99Onkuyz
19808
tnSmYItJf5UE8UpdglsoQCyaOfKCp/g9HxU4wRBM87fIr2vBvHyyQqAEVH7f73PrqJ+3+Dxq7Qzd
19809
4wfXcNguXClOKe7Ankf7GoyD7BW4SpKCtWrW2PJRdIfiqUUhp18ZD7IiD+XPUVnJKEYkSOWDkHwg
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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19949
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19966
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19967
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19984
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19985
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19989
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19990
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19992
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19994
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19998
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20000
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20028
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20111
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20112
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20119
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20120
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20121
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20122
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20123
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20124
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20133
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20136
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20139
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20141
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20202
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20203
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''',
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''',
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'hamming10-4' : '''
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''',
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''',
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30870
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30871
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30877
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33818
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33820
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33824
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33832
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33833
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33839
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33880
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41403
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41405
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41407
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41409
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41410
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41411
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41412
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41413
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41414
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41415
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41416
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41417
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41418
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41419
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41420
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41421
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41422
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41423
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41424
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41425
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41426
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41427
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41428
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41429
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41430
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41431
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41432
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41433
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41434
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41435
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41436
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41437
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41438
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41439
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41440
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41441
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41442
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41443
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41444
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41445
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41446
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41447
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41448
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41449
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41450
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41451
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41452
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41453
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41454
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41455
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41456
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41457
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41458
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41459
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41460
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41461
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41462
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41463
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41467
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41468
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41470
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41471
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41483
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41485
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41489
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41495
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41500
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41501
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41502
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41503
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41505
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41506
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41508
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41509
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41510
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41511
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41513
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41514
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41941
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41942
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41943
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42000
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42014
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42019
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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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43381
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43384
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43385
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''',
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'gen200_p0.9_55' : '''
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''',
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'p_hat1500-2' : '''
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50671
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50673
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50674
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50680
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50683
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50684
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50689
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50690
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50691
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50692
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50693
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50694
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50695
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50696
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50698
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50699
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50700
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50707
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50708
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50709
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50710
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50737
D12zT13z2wIA
50738
'''}
50739
from cStringIO import StringIO
50740
from base64 import b64decode
50741
from gzip import GzipFile
50742
50743
for name in dimacs_g6_base64_dict:
50744
f = GzipFile(mode='r', fileobj=StringIO(b64decode(dimacs_g6_base64_dict[name])))
50745
g6string = f.read()
50746
f.close()
50747
g = Graph(g6string)
50748
g.name(new = name)
50749
add_to_lists(g, dimacs_graphs, all_graphs)
50750
print "loaded graph - ", g
50751
50752
print("loaded graphs")
50753
50754
#############################################################################
50755
# End of graphs section #
50756
#############################################################################
50757
50758
print("\nRemember to load DIMACS and Sloane graphs if you want them")
50759
50760
#############################################################################
50761
# End of gt.sage file #
50762
#############################################################################
50763
50764