Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/site-packages/networkx/algorithms/asteroidal.py @ 0:4f3585e2f14b draft default tip
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author | shellac |
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date | Mon, 22 Mar 2021 18:12:50 +0000 |
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1 """ |
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2 Algorithms for asteroidal triples and asteroidal numbers in graphs. |
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3 |
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4 An asteroidal triple in a graph G is a set of three non-adjacent vertices |
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5 u, v and w such that there exist a path between any two of them that avoids |
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6 closed neighborhood of the third. More formally, v_j, v_k belongs to the same |
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7 connected component of G - N[v_i], where N[v_i] denotes the closed neighborhood |
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8 of v_i. A graph which does not contain any asteroidal triples is called |
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9 an AT-free graph. The class of AT-free graphs is a graph class for which |
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10 many NP-complete problems are solvable in polynomial time. Amongst them, |
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11 independent set and coloring. |
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12 """ |
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13 import networkx as nx |
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14 from networkx.utils import not_implemented_for |
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15 |
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16 __all__ = ["is_at_free", "find_asteroidal_triple"] |
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17 |
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18 |
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19 @not_implemented_for("directed") |
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20 @not_implemented_for("multigraph") |
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21 def find_asteroidal_triple(G): |
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22 r"""Find an asteroidal triple in the given graph. |
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23 |
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24 An asteroidal triple is a triple of non-adjacent vertices such that |
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25 there exists a path between any two of them which avoids the closed |
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26 neighborhood of the third. It checks all independent triples of vertices |
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27 and whether they are an asteroidal triple or not. This is done with the |
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28 help of a data structure called a component structure. |
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29 A component structure encodes information about which vertices belongs to |
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30 the same connected component when the closed neighborhood of a given vertex |
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31 is removed from the graph. The algorithm used to check is the trivial |
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32 one, outlined in [1]_, which has a runtime of |
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33 :math:`O(|V||\overline{E} + |V||E|)`, where the second term is the |
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34 creation of the component structure. |
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35 |
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36 Parameters |
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37 ---------- |
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38 G : NetworkX Graph |
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39 The graph to check whether is AT-free or not |
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40 |
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41 Returns |
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42 ------- |
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43 list or None |
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44 An asteroidal triple is returned as a list of nodes. If no asteroidal |
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45 triple exists, i.e. the graph is AT-free, then None is returned. |
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46 The returned value depends on the certificate parameter. The default |
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47 option is a bool which is True if the graph is AT-free, i.e. the |
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48 given graph contains no asteroidal triples, and False otherwise, i.e. |
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49 if the graph contains at least one asteroidal triple. |
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50 |
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51 Notes |
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52 ----- |
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53 The component structure and the algorithm is described in [1]_. The current |
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54 implementation implements the trivial algorithm for simple graphs. |
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55 |
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56 References |
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57 ---------- |
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58 .. [1] Ekkehard Köhler, |
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59 "Recognizing Graphs without asteroidal triples", |
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60 Journal of Discrete Algorithms 2, pages 439-452, 2004. |
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61 https://www.sciencedirect.com/science/article/pii/S157086670400019X |
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62 """ |
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63 V = set(G.nodes) |
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64 |
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65 if len(V) < 6: |
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66 # An asteroidal triple cannot exist in a graph with 5 or less vertices. |
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67 return None |
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68 |
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69 component_structure = create_component_structure(G) |
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70 E_complement = set(nx.complement(G).edges) |
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71 |
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72 for e in E_complement: |
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73 u = e[0] |
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74 v = e[1] |
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75 u_neighborhood = set(G[u]).union([u]) |
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76 v_neighborhood = set(G[v]).union([v]) |
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77 union_of_neighborhoods = u_neighborhood.union(v_neighborhood) |
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78 for w in V - union_of_neighborhoods: |
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79 """Check for each pair of vertices whether they belong to the |
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80 same connected component when the closed neighborhood of the |
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81 third is removed.""" |
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82 if ( |
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83 component_structure[u][v] == component_structure[u][w] |
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84 and component_structure[v][u] == component_structure[v][w] |
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85 and component_structure[w][u] == component_structure[w][v] |
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86 ): |
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87 return [u, v, w] |
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88 |
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89 return None |
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90 |
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91 |
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92 @not_implemented_for("directed") |
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93 @not_implemented_for("multigraph") |
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94 def is_at_free(G): |
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95 """Check if a graph is AT-free. |
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96 |
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97 The method uses the `find_asteroidal_triple` method to recognize |
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98 an AT-free graph. If no asteroidal triple is found the graph is |
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99 AT-free and True is returned. If at least one asteroidal triple is |
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100 found the graph is not AT-free and False is returned. |
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101 |
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102 Parameters |
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103 ---------- |
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104 G : NetworkX Graph |
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105 The graph to check whether is AT-free or not. |
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106 |
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107 Returns |
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108 ------- |
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109 bool |
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110 True if G is AT-free and False otherwise. |
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111 |
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112 Examples |
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113 -------- |
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114 >>> G = nx.Graph([(0, 1), (0, 2), (1, 2), (1, 3), (1, 4), (4, 5)]) |
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115 >>> nx.is_at_free(G) |
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116 True |
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117 |
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118 >>> G = nx.cycle_graph(6) |
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119 >>> nx.is_at_free(G) |
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120 False |
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121 """ |
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122 return find_asteroidal_triple(G) is None |
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123 |
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124 |
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125 @not_implemented_for("directed") |
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126 @not_implemented_for("multigraph") |
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127 def create_component_structure(G): |
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128 r"""Create component structure for G. |
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129 |
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130 A *component structure* is an `nxn` array, denoted `c`, where `n` is |
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131 the number of vertices, where each row and column corresponds to a vertex. |
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132 |
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133 .. math:: |
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134 c_{uv} = \begin{cases} 0, if v \in N[u] \\ |
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135 k, if v \in component k of G \setminus N[u] \end{cases} |
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136 |
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137 Where `k` is an arbitrary label for each component. The structure is used |
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138 to simplify the detection of asteroidal triples. |
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139 |
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140 Parameters |
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141 ---------- |
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142 G : NetworkX Graph |
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143 Undirected, simple graph. |
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144 |
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145 Returns |
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146 ------- |
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147 component_structure : dictionary |
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148 A dictionary of dictionaries, keyed by pairs of vertices. |
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149 |
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150 """ |
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151 V = set(G.nodes) |
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152 component_structure = {} |
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153 for v in V: |
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154 label = 0 |
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155 closed_neighborhood = set(G[v]).union({v}) |
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156 row_dict = {} |
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157 for u in closed_neighborhood: |
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158 row_dict[u] = 0 |
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159 |
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160 G_reduced = G.subgraph(set(G.nodes) - closed_neighborhood) |
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161 for cc in nx.connected_components(G_reduced): |
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162 label += 1 |
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163 for u in cc: |
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164 row_dict[u] = label |
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165 |
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166 component_structure[v] = row_dict |
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167 |
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168 return component_structure |