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annotate planemo/lib/python3.7/site-packages/networkx/generators/line.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 # Copyright (C) 2013-2019 by
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2 # Aric Hagberg <hagberg@lanl.gov>
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3 # Dan Schult <dschult@colgate.edu>
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4 # Pieter Swart <swart@lanl.gov>
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5 # All rights reserved.
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6 # BSD license.
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7 #
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8 # Authors: James Clough <james.clough91@gmail.com>
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9 # Aric Hagberg <hagberg@lanl.gov>
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10 # Pieter Swart <swart@lanl.gov>
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11 # Dan Schult <dschult@colgate.edu>
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12 # chebee7i <chebee7i@gmail.com>
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13 """Functions for generating line graphs."""
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14 from itertools import combinations
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15 from collections import defaultdict
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16
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17 import networkx as nx
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18 from networkx.utils import arbitrary_element, generate_unique_node
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19 from networkx.utils.decorators import *
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20
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21 __all__ = ['line_graph', 'inverse_line_graph']
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22
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23
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24 def line_graph(G, create_using=None):
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25 r"""Returns the line graph of the graph or digraph `G`.
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26
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27 The line graph of a graph `G` has a node for each edge in `G` and an
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28 edge joining those nodes if the two edges in `G` share a common node. For
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29 directed graphs, nodes are adjacent exactly when the edges they represent
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30 form a directed path of length two.
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31
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32 The nodes of the line graph are 2-tuples of nodes in the original graph (or
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33 3-tuples for multigraphs, with the key of the edge as the third element).
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34
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35 For information about self-loops and more discussion, see the **Notes**
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36 section below.
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37
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38 Parameters
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39 ----------
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40 G : graph
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41 A NetworkX Graph, DiGraph, MultiGraph, or MultiDigraph.
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42 create_using : NetworkX graph constructor, optional (default=nx.Graph)
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43 Graph type to create. If graph instance, then cleared before populated.
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44
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45 Returns
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46 -------
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47 L : graph
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48 The line graph of G.
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49
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50 Examples
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51 --------
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52 >>> import networkx as nx
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53 >>> G = nx.star_graph(3)
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54 >>> L = nx.line_graph(G)
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55 >>> print(sorted(map(sorted, L.edges()))) # makes a 3-clique, K3
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56 [[(0, 1), (0, 2)], [(0, 1), (0, 3)], [(0, 2), (0, 3)]]
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57
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58 Notes
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59 -----
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60 Graph, node, and edge data are not propagated to the new graph. For
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61 undirected graphs, the nodes in G must be sortable, otherwise the
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62 constructed line graph may not be correct.
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63
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64 *Self-loops in undirected graphs*
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65
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66 For an undirected graph `G` without multiple edges, each edge can be
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67 written as a set `\{u, v\}`. Its line graph `L` has the edges of `G` as
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68 its nodes. If `x` and `y` are two nodes in `L`, then `\{x, y\}` is an edge
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69 in `L` if and only if the intersection of `x` and `y` is nonempty. Thus,
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70 the set of all edges is determined by the set of all pairwise intersections
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71 of edges in `G`.
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72
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73 Trivially, every edge in G would have a nonzero intersection with itself,
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74 and so every node in `L` should have a self-loop. This is not so
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75 interesting, and the original context of line graphs was with simple
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76 graphs, which had no self-loops or multiple edges. The line graph was also
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77 meant to be a simple graph and thus, self-loops in `L` are not part of the
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78 standard definition of a line graph. In a pairwise intersection matrix,
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79 this is analogous to excluding the diagonal entries from the line graph
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80 definition.
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81
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82 Self-loops and multiple edges in `G` add nodes to `L` in a natural way, and
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83 do not require any fundamental changes to the definition. It might be
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84 argued that the self-loops we excluded before should now be included.
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85 However, the self-loops are still "trivial" in some sense and thus, are
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86 usually excluded.
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87
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88 *Self-loops in directed graphs*
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89
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90 For a directed graph `G` without multiple edges, each edge can be written
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91 as a tuple `(u, v)`. Its line graph `L` has the edges of `G` as its
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92 nodes. If `x` and `y` are two nodes in `L`, then `(x, y)` is an edge in `L`
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93 if and only if the tail of `x` matches the head of `y`, for example, if `x
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94 = (a, b)` and `y = (b, c)` for some vertices `a`, `b`, and `c` in `G`.
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95
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96 Due to the directed nature of the edges, it is no longer the case that
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97 every edge in `G` should have a self-loop in `L`. Now, the only time
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98 self-loops arise is if a node in `G` itself has a self-loop. So such
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99 self-loops are no longer "trivial" but instead, represent essential
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100 features of the topology of `G`. For this reason, the historical
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101 development of line digraphs is such that self-loops are included. When the
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102 graph `G` has multiple edges, once again only superficial changes are
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103 required to the definition.
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104
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105 References
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106 ----------
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107 * Harary, Frank, and Norman, Robert Z., "Some properties of line digraphs",
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108 Rend. Circ. Mat. Palermo, II. Ser. 9 (1960), 161--168.
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109 * Hemminger, R. L.; Beineke, L. W. (1978), "Line graphs and line digraphs",
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110 in Beineke, L. W.; Wilson, R. J., Selected Topics in Graph Theory,
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111 Academic Press Inc., pp. 271--305.
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112
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113 """
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114 if G.is_directed():
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115 L = _lg_directed(G, create_using=create_using)
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116 else:
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117 L = _lg_undirected(G, selfloops=False, create_using=create_using)
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118 return L
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119
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120
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121 def _node_func(G):
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122 """Returns a function which returns a sorted node for line graphs.
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123
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124 When constructing a line graph for undirected graphs, we must normalize
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125 the ordering of nodes as they appear in the edge.
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126
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127 """
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128 if G.is_multigraph():
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129 def sorted_node(u, v, key):
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130 return (u, v, key) if u <= v else (v, u, key)
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131 else:
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132 def sorted_node(u, v):
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133 return (u, v) if u <= v else (v, u)
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134 return sorted_node
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135
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136
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137 def _edge_func(G):
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138 """Returns the edges from G, handling keys for multigraphs as necessary.
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139
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140 """
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141 if G.is_multigraph():
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142 def get_edges(nbunch=None):
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143 return G.edges(nbunch, keys=True)
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144 else:
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145 def get_edges(nbunch=None):
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146 return G.edges(nbunch)
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147 return get_edges
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148
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149
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150 def _sorted_edge(u, v):
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151 """Returns a sorted edge.
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152
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153 During the construction of a line graph for undirected graphs, the data
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154 structure can be a multigraph even though the line graph will never have
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155 multiple edges between its nodes. For this reason, we must make sure not
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156 to add any edge more than once. This requires that we build up a list of
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157 edges to add and then remove all duplicates. And so, we must normalize
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158 the representation of the edges.
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159
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160 """
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161 return (u, v) if u <= v else (v, u)
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162
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163
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164 def _lg_directed(G, create_using=None):
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165 """Returns the line graph L of the (multi)digraph G.
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166
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167 Edges in G appear as nodes in L, represented as tuples of the form (u,v)
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168 or (u,v,key) if G is a multidigraph. A node in L corresponding to the edge
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169 (u,v) is connected to every node corresponding to an edge (v,w).
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170
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171 Parameters
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172 ----------
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173 G : digraph
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174 A directed graph or directed multigraph.
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175 create_using : NetworkX graph constructor, optional
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176 Graph type to create. If graph instance, then cleared before populated.
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177 Default is to use the same graph class as `G`.
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178
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179 """
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180 L = nx.empty_graph(0, create_using, default=G.__class__)
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181
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182 # Create a graph specific edge function.
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183 get_edges = _edge_func(G)
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184
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185 for from_node in get_edges():
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186 # from_node is: (u,v) or (u,v,key)
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187 L.add_node(from_node)
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188 for to_node in get_edges(from_node[1]):
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189 L.add_edge(from_node, to_node)
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190
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191 return L
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192
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193
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194 def _lg_undirected(G, selfloops=False, create_using=None):
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195 """Returns the line graph L of the (multi)graph G.
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196
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197 Edges in G appear as nodes in L, represented as sorted tuples of the form
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198 (u,v), or (u,v,key) if G is a multigraph. A node in L corresponding to
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199 the edge {u,v} is connected to every node corresponding to an edge that
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200 involves u or v.
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201
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202 Parameters
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203 ----------
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204 G : graph
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205 An undirected graph or multigraph.
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206 selfloops : bool
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207 If `True`, then self-loops are included in the line graph. If `False`,
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208 they are excluded.
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209 create_using : NetworkX graph constructor, optional (default=nx.Graph)
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210 Graph type to create. If graph instance, then cleared before populated.
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211
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212 Notes
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213 -----
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214 The standard algorithm for line graphs of undirected graphs does not
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215 produce self-loops.
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216
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217 """
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218 L = nx.empty_graph(0, create_using, default=G.__class__)
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219
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220 # Graph specific functions for edges and sorted nodes.
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221 get_edges = _edge_func(G)
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222 sorted_node = _node_func(G)
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223
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224 # Determine if we include self-loops or not.
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225 shift = 0 if selfloops else 1
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226
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227 edges = set([])
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228 for u in G:
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229 # Label nodes as a sorted tuple of nodes in original graph.
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230 nodes = [sorted_node(*x) for x in get_edges(u)]
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231
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232 if len(nodes) == 1:
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233 # Then the edge will be an isolated node in L.
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234 L.add_node(nodes[0])
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235
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236 # Add a clique of `nodes` to graph. To prevent double adding edges,
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237 # especially important for multigraphs, we store the edges in
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238 # canonical form in a set.
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239 for i, a in enumerate(nodes):
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240 edges.update([_sorted_edge(a, b) for b in nodes[i + shift:]])
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241
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242 L.add_edges_from(edges)
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243 return L
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244
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245
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246 @not_implemented_for('directed')
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247 @not_implemented_for('multigraph')
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248 def inverse_line_graph(G):
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249 """ Returns the inverse line graph of graph G.
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250
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251 If H is a graph, and G is the line graph of H, such that H = L(G).
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252 Then H is the inverse line graph of G.
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253
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254 Not all graphs are line graphs and these do not have an inverse line graph.
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255 In these cases this generator returns a NetworkXError.
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256
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257 Parameters
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258 ----------
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259 G : graph
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260 A NetworkX Graph
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261
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262 Returns
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263 -------
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264 H : graph
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265 The inverse line graph of G.
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266
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267 Raises
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268 ------
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269 NetworkXNotImplemented
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270 If G is directed or a multigraph
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271
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272 NetworkXError
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273 If G is not a line graph
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274
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275 Notes
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276 -----
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277 This is an implementation of the Roussopoulos algorithm.
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278
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279 If G consists of multiple components, then the algorithm doesn't work.
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280 You should invert every component seperately:
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281
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282 >>> K5 = nx.complete_graph(5)
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283 >>> P4 = nx.Graph([('a', 'b'), ('b', 'c'), ('c', 'd')])
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284 >>> G = nx.union(K5, P4)
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285 >>> root_graphs = []
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286 >>> for comp in nx.connected_components(G):
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287 ... root_graphs.append(nx.inverse_line_graph(G.subgraph(comp)))
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288 >>> len(root_graphs)
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289 2
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290
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291 References
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292 ----------
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293 * Roussopolous, N, "A max {m, n} algorithm for determining the graph H from
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294 its line graph G", Information Processing Letters 2, (1973), 108--112.
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295
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296 """
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297 if G.number_of_nodes() == 0:
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298 a = generate_unique_node()
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299 H = nx.Graph()
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300 H.add_node(a)
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301 return H
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302 elif G.number_of_nodes() == 1:
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303 v = list(G)[0]
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304 a = (v, 0)
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305 b = (v, 1)
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306 H = nx.Graph([(a,b)])
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307 return H
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308 elif G.number_of_nodes() > 1 and G.number_of_edges() == 0:
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309 msg = (
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310 "inverse_line_graph() doesn't work on an edgeless graph. "
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311 "Please use this function on each component seperately."
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312 )
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313 raise nx.NetworkXError(msg)
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314
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315 starting_cell = _select_starting_cell(G)
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316 P = _find_partition(G, starting_cell)
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317 # count how many times each vertex appears in the partition set
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318 P_count = {u: 0 for u in G.nodes()}
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319 for p in P:
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320 for u in p:
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321 P_count[u] += 1
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322
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323 if max(P_count.values()) > 2:
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324 msg = "G is not a line graph (vertex found in more " \
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325 "than two partition cells)"
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326 raise nx.NetworkXError(msg)
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327 W = tuple([(u,) for u in P_count if P_count[u] == 1])
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328 H = nx.Graph()
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329 H.add_nodes_from(P)
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330 H.add_nodes_from(W)
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331 for a, b in combinations(H.nodes(), 2):
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332 if len(set(a).intersection(set(b))) > 0:
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333 H.add_edge(a, b)
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334 return H
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335
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336
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337 def _triangles(G, e):
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338 """ Return list of all triangles containing edge e"""
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339 u, v = e
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340 if u not in G:
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341 raise nx.NetworkXError("Vertex %s not in graph" % u)
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342 if v not in G[u]:
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343 raise nx.NetworkXError("Edge (%s, %s) not in graph" % (u, v))
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344 triangle_list = []
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345 for x in G[u]:
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346 if x in G[v]:
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347 triangle_list.append((u, v, x))
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348 return triangle_list
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349
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350
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351 def _odd_triangle(G, T):
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352 """ Test whether T is an odd triangle in G
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353
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354 Parameters
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355 ----------
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356 G : NetworkX Graph
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357 T : 3-tuple of vertices forming triangle in G
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358
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359 Returns
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360 -------
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361 True is T is an odd triangle
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362 False otherwise
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363
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364 Raises
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365 ------
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366 NetworkXError
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367 T is not a triangle in G
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368
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369 Notes
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370 -----
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371 An odd triangle is one in which there exists another vertex in G which is
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372 adjacent to either exactly one or exactly all three of the vertices in the
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373 triangle.
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374
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375 """
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376 for u in T:
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377 if u not in G.nodes():
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378 raise nx.NetworkXError("Vertex %s not in graph" % u)
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379 for e in list(combinations(T, 2)):
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380 if e[0] not in G[e[1]]:
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381 raise nx.NetworkXError("Edge (%s, %s) not in graph" % (e[0], e[1]))
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382
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383 T_neighbors = defaultdict(int)
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384 for t in T:
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385 for v in G[t]:
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diff changeset
386 if v not in T:
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387 T_neighbors[v] += 1
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parents:
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388 for v in T_neighbors:
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389 if T_neighbors[v] in [1, 3]:
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390 return True
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391 return False
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392
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393
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394 def _find_partition(G, starting_cell):
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diff changeset
395 """ Find a partition of the vertices of G into cells of complete graphs
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396
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397 Parameters
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398 ----------
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399 G : NetworkX Graph
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400 starting_cell : tuple of vertices in G which form a cell
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401
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402 Returns
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403 -------
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404 List of tuples of vertices of G
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405
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406 Raises
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407 ------
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408 NetworkXError
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409 If a cell is not a complete subgraph then G is not a line graph
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410 """
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411 G_partition = G.copy()
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412 P = [starting_cell] # partition set
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413 G_partition.remove_edges_from(list(combinations(starting_cell, 2)))
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414 # keep list of partitioned nodes which might have an edge in G_partition
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415 partitioned_vertices = list(starting_cell)
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416 while G_partition.number_of_edges() > 0:
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417 # there are still edges left and so more cells to be made
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418 u = partitioned_vertices[-1]
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419 deg_u = len(G_partition[u])
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420 if deg_u == 0:
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421 # if u has no edges left in G_partition then we have found
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422 # all of its cells so we do not need to keep looking
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423 partitioned_vertices.pop()
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424 else:
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diff changeset
425 # if u still has edges then we need to find its other cell
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426 # this other cell must be a complete subgraph or else G is
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427 # not a line graph
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428 new_cell = [u] + list(G_partition[u])
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429 for u in new_cell:
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430 for v in new_cell:
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431 if (u != v) and (v not in G_partition[u]):
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432 msg = "G is not a line graph" \
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433 "(partition cell not a complete subgraph)"
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434 raise nx.NetworkXError(msg)
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435 P.append(tuple(new_cell))
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diff changeset
436 G_partition.remove_edges_from(list(combinations(new_cell, 2)))
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437 partitioned_vertices += new_cell
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438 return P
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439
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440
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441 def _select_starting_cell(G, starting_edge=None):
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442 """ Select a cell to initiate _find_partition
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443
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444 Parameters
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445 ----------
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446 G : NetworkX Graph
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447 starting_edge: an edge to build the starting cell from
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448
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449 Returns
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450 -------
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451 Tuple of vertices in G
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452
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453 Raises
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454 ------
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455 NetworkXError
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456 If it is determined that G is not a line graph
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457
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458 Notes
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459 -----
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460 If starting edge not specified then pick an arbitrary edge - doesn't
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461 matter which. However, this function may call itself requiring a
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462 specific starting edge. Note that the r, s notation for counting
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diff changeset
463 triangles is the same as in the Roussopoulos paper cited above.
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464 """
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465 if starting_edge is None:
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466 e = arbitrary_element(list(G.edges()))
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467 else:
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468 e = starting_edge
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469 if e[0] not in G[e[1]]:
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470 msg = 'starting_edge (%s, %s) is not in the Graph'
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471 raise nx.NetworkXError(msg % e)
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472 e_triangles = _triangles(G, e)
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473 r = len(e_triangles)
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474 if r == 0:
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parents:
diff changeset
475 # there are no triangles containing e, so the starting cell is just e
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476 starting_cell = e
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477 elif r == 1:
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parents:
diff changeset
478 # there is exactly one triangle, T, containing e. If other 2 edges
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diff changeset
479 # of T belong only to this triangle then T is starting cell
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480 T = e_triangles[0]
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diff changeset
481 a, b, c = T
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parents:
diff changeset
482 # ab was original edge so check the other 2 edges
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diff changeset
483 ac_edges = [x for x in _triangles(G, (a, c))]
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diff changeset
484 bc_edges = [x for x in _triangles(G, (b, c))]
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485 if len(ac_edges) == 1:
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diff changeset
486 if len(bc_edges) == 1:
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diff changeset
487 starting_cell = T
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488 else:
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489 return _select_starting_cell(G, starting_edge=(b, c))
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490 else:
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diff changeset
491 return _select_starting_cell(G, starting_edge=(a, c))
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492 else:
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diff changeset
493 # r >= 2 so we need to count the number of odd triangles, s
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diff changeset
494 s = 0
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495 odd_triangles = []
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496 for T in e_triangles:
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497 if _odd_triangle(G, T):
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498 s += 1
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499 odd_triangles.append(T)
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parents:
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500 if r == 2 and s == 0:
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parents:
diff changeset
501 # in this case either triangle works, so just use T
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502 starting_cell = T
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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parents:
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503 elif r - 1 <= s <= r:
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guerler
parents:
diff changeset
504 # check if odd triangles containing e form complete subgraph
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505 # there must be exactly s+2 of them
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parents:
diff changeset
506 # and they must all be connected
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507 triangle_nodes = set([])
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parents:
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508 for T in odd_triangles:
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guerler
parents:
diff changeset
509 for x in T:
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parents:
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510 triangle_nodes.add(x)
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guerler
parents:
diff changeset
511 if len(triangle_nodes) == s + 2:
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guerler
parents:
diff changeset
512 for u in triangle_nodes:
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guerler
parents:
diff changeset
513 for v in triangle_nodes:
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
514 if u != v and (v not in G[u]):
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
515 msg = "G is not a line graph (odd triangles " \
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
516 "do not form complete subgraph)"
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
517 raise nx.NetworkXError(msg)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
518 # otherwise then we can use this as the starting cell
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
519 starting_cell = tuple(triangle_nodes)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
520 else:
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
521 msg = "G is not a line graph (odd triangles " \
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
522 "do not form complete subgraph)"
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
523 raise nx.NetworkXError(msg)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
524 else:
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
525 msg = "G is not a line graph (incorrect number of " \
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
526 "odd triangles around starting edge)"
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
527 raise nx.NetworkXError(msg)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
528 return starting_cell