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annotate planemo/lib/python3.7/site-packages/networkx/algorithms/covering.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 2016-2019 NetworkX developers.
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2 # Copyright (C) 2016 by
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3 # Nishant Nikhil <nishantiam@gmail.com>
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4 # All rights reserved.
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5 # BSD license.
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6
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7 """ Functions related to graph covers."""
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8
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9 import networkx as nx
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10 from networkx.utils import not_implemented_for, arbitrary_element
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11 from functools import partial
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12 from itertools import chain
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13
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14
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15 __all__ = ['min_edge_cover', 'is_edge_cover']
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16
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17
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18 @not_implemented_for('directed')
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19 @not_implemented_for('multigraph')
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20 def min_edge_cover(G, matching_algorithm=None):
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21 """Returns a set of edges which constitutes
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22 the minimum edge cover of the graph.
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23
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24 A smallest edge cover can be found in polynomial time by finding
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25 a maximum matching and extending it greedily so that all nodes
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26 are covered.
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27
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28 Parameters
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29 ----------
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30 G : NetworkX graph
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31 An undirected bipartite graph.
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32
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33 matching_algorithm : function
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34 A function that returns a maximum cardinality matching in a
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35 given bipartite graph. The function must take one input, the
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36 graph ``G``, and return a dictionary mapping each node to its
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37 mate. If not specified,
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38 :func:`~networkx.algorithms.bipartite.matching.hopcroft_karp_matching`
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39 will be used. Other possibilities include
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40 :func:`~networkx.algorithms.bipartite.matching.eppstein_matching`,
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41 or matching algorithms in the
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42 :mod:`networkx.algorithms.matching` module.
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43
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44 Returns
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45 -------
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46 min_cover : set
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47
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48 It contains all the edges of minimum edge cover
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49 in form of tuples. It contains both the edges `(u, v)` and `(v, u)`
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50 for given nodes `u` and `v` among the edges of minimum edge cover.
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51
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52 Notes
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53 -----
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54 An edge cover of a graph is a set of edges such that every node of
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55 the graph is incident to at least one edge of the set.
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56 The minimum edge cover is an edge covering of smallest cardinality.
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57
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58 Due to its implementation, the worst-case running time of this algorithm
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59 is bounded by the worst-case running time of the function
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60 ``matching_algorithm``.
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61
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62 Minimum edge cover for bipartite graph can also be found using the
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63 function present in :mod:`networkx.algorithms.bipartite.covering`
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64 """
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65 if nx.number_of_isolates(G) > 0:
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66 # ``min_cover`` does not exist as there is an isolated node
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67 raise nx.NetworkXException(
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68 "Graph has a node with no edge incident on it, "
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69 "so no edge cover exists.")
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70 if matching_algorithm is None:
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71 matching_algorithm = partial(nx.max_weight_matching,
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72 maxcardinality=True)
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73 maximum_matching = matching_algorithm(G)
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74 # ``min_cover`` is superset of ``maximum_matching``
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75 try:
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76 min_cover = set(maximum_matching.items()) # bipartite matching case returns dict
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77 except AttributeError:
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78 min_cover = maximum_matching
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79 # iterate for uncovered nodes
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80 uncovered_nodes = set(G) - {v for u, v in min_cover} - {u for u, v in min_cover}
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81 for v in uncovered_nodes:
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82 # Since `v` is uncovered, each edge incident to `v` will join it
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83 # with a covered node (otherwise, if there were an edge joining
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84 # uncovered nodes `u` and `v`, the maximum matching algorithm
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85 # would have found it), so we can choose an arbitrary edge
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86 # incident to `v`. (This applies only in a simple graph, not a
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87 # multigraph.)
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88 u = arbitrary_element(G[v])
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89 min_cover.add((u, v))
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90 min_cover.add((v, u))
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91 return min_cover
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92
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93
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94 @not_implemented_for('directed')
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95 def is_edge_cover(G, cover):
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96 """Decides whether a set of edges is a valid edge cover of the graph.
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97
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98 Given a set of edges, whether it is an edge covering can
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99 be decided if we just check whether all nodes of the graph
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100 has an edge from the set, incident on it.
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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 An undirected bipartite graph.
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106
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107 cover : set
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108 Set of edges to be checked.
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109
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110 Returns
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111 -------
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112 bool
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113 Whether the set of edges is a valid edge cover of the graph.
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114
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115 Notes
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116 -----
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117 An edge cover of a graph is a set of edges such that every node of
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118 the graph is incident to at least one edge of the set.
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119 """
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120 return set(G) <= set(chain.from_iterable(cover))