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annotate planemo/lib/python3.7/site-packages/networkx/algorithms/asteroidal.py @ 1:56ad4e20f292 draft

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"planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
author guerler
date Fri, 31 Jul 2020 00:32:28 -0400
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1 # -*- coding: utf-8 -*-
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2 # Copyright (C) 2004-2019 by
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3 # Aric Hagberg <hagberg@lanl.gov>
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4 # Dan Schult <dschult@colgate.edu>
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5 # Pieter Swart <swart@lanl.gov>
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6 # All rights reserved.
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7 # BSD license.
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8 #
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9 # Authors: Haakon H. Rød (haakonhr@gmail.com)
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10 """
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11 Algorithms for asteroidal triples and asteroidal numbers in graphs.
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12
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13 An asteroidal triple in a graph G is a set of three non-adjacent vertices
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14 u, v and w such that there exist a path between any two of them that avoids
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15 closed neighborhood of the third. More formally, v_j, v_k belongs to the same
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16 connected component of G - N[v_i], where N[v_i] denotes the closed neighborhood
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17 of v_i. A graph which does not contain any asteroidal triples is called
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18 an AT-free graph. The class of AT-free graphs is a graph class for which
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19 many NP-complete problems are solvable in polynomial time. Amongst them,
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20 independent set and coloring.
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21 """
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22 import networkx as nx
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23 from networkx.utils import not_implemented_for
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24
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25 __all__ = ["is_at_free", "find_asteroidal_triple"]
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26
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27
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28 @not_implemented_for("directed")
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29 @not_implemented_for("multigraph")
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30 def find_asteroidal_triple(G):
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31 r"""Find an asteroidal triple in the given graph.
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32
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33 An asteroidal triple is a triple of non-adjacent vertices such that
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34 there exists a path between any two of them which avoids the closed
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35 neighborhood of the third. It checks all independent triples of vertices
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36 and whether they are an asteroidal triple or not. This is done with the
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37 help of a data structure called a component structure.
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38 A component structure encodes information about which vertices belongs to
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39 the same connected component when the closed neighborhood of a given vertex
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40 is removed from the graph. The algorithm used to check is the trivial
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41 one, outlined in [1]_, which has a runtime of
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42 :math:`O(|V||\overline{E} + |V||E|)`, where the second term is the
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43 creation of the component structure.
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44
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45 Parameters
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46 ----------
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47 G : NetworkX Graph
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48 The graph to check whether is AT-free or not
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49
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50 Returns
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51 -------
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52 list or None
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53 An asteroidal triple is returned as a list of nodes. If no asteroidal
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54 triple exists, i.e. the graph is AT-free, then None is returned.
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55 The returned value depends on the certificate parameter. The default
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56 option is a bool which is True if the graph is AT-free, i.e. the
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57 given graph contains no asteroidal triples, and False otherwise, i.e.
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58 if the graph contains at least one asteroidal triple.
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59
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60 Notes
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61 -----
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62 The component structure and the algorithm is described in [1]_. The current
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63 implementation implements the trivial algorithm for simple graphs.
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64
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65 References
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66 ----------
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67 .. [1] Ekkehard Köhler,
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68 "Recognizing Graphs without asteroidal triples",
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69 Journal of Discrete Algorithms 2, pages 439-452, 2004.
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70 https://www.sciencedirect.com/science/article/pii/S157086670400019X
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71 """
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72 V = set(G.nodes)
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73
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74 if len(V) < 6:
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75 # An asteroidal triple cannot exist in a graph with 5 or less vertices.
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76 return None
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77
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78 component_structure = create_component_structure(G)
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79 E_complement = set(nx.complement(G).edges)
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80
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81 for e in E_complement:
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82 u = e[0]
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83 v = e[1]
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84 u_neighborhood = set(G[u]).union([u])
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85 v_neighborhood = set(G[v]).union([v])
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86 union_of_neighborhoods = u_neighborhood.union(v_neighborhood)
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87 for w in V - union_of_neighborhoods:
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88 """Check for each pair of vertices whether they belong to the
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89 same connected component when the closed neighborhood of the
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90 third is removed."""
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91 if (component_structure[u][v] == component_structure[u][w] and
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92 component_structure[v][u] == component_structure[v][w] and
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93 component_structure[w][u] == component_structure[w][v]):
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94 return [u, v, w]
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95
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96 return None
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97
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98
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99 @not_implemented_for("directed")
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100 @not_implemented_for("multigraph")
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101 def is_at_free(G):
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102 """Check if a graph is AT-free.
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103
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104 The method uses the `find_asteroidal_triple` method to recognize
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105 an AT-free graph. If no asteroidal triple is found the graph is
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106 AT-free and True is returned. If at least one asteroidal triple is
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107 found the graph is not AT-free and False is returned.
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108
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109 Parameters
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110 ----------
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111 G : NetworkX Graph
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112 The graph to check whether is AT-free or not.
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113
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114 Returns
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115 -------
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116 bool
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117 True if G is AT-free and False otherwise.
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118
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119 Examples
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120 --------
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121 >>> G = nx.Graph([(0, 1), (0, 2), (1, 2), (1, 3), (1, 4), (4, 5)])
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122 >>> nx.is_at_free(G)
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123 True
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124
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125 >>> G = nx.cycle_graph(6)
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126 >>> nx.is_at_free(G)
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127 False
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128 """
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129 return find_asteroidal_triple(G) is None
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130
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131
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132 @not_implemented_for("directed")
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133 @not_implemented_for("multigraph")
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134 def create_component_structure(G):
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135 r"""Create component structure for G.
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136
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137 A *component structure* is an `nxn` array, denoted `c`, where `n` is
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138 the number of vertices, where each row and column corresponds to a vertex.
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139
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140 .. math::
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141 c_{uv} = \begin{cases} 0, if v \in N[u] \\
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142 k, if v \in component k of G \setminus N[u] \end{cases}
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143
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144 Where `k` is an arbitrary label for each component. The structure is used
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145 to simplify the detection of asteroidal triples.
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146
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147 Parameters
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148 ----------
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149 G : NetworkX Graph
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150 Undirected, simple graph.
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151
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152 Returns
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153 -------
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154 component_structure : dictionary
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155 A dictionary of dictionaries, keyed by pairs of vertices.
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156
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157 """
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158 V = set(G.nodes)
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159 component_structure = {}
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160 for v in V:
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161 label = 0
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162 closed_neighborhood = set(G[v]).union(set([v]))
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163 row_dict = {}
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164 for u in closed_neighborhood:
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165 row_dict[u] = 0
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166
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167 G_reduced = G.subgraph(set(G.nodes) - closed_neighborhood)
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168 for cc in nx.connected_components(G_reduced):
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169 label += 1
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170 for u in cc:
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171 row_dict[u] = label
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172
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173 component_structure[v] = row_dict
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174
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175 return component_structure