Warning

This documents an unmaintained version of NetworkX. Please upgrade to a maintained version and see the current NetworkX documentation.

# Source code for networkx.algorithms.covering

#    Copyright 2016-2019 NetworkX developers.
#    Nishant Nikhil <nishantiam@gmail.com>

""" Functions related to graph covers."""

import networkx as nx
from networkx.utils import not_implemented_for, arbitrary_element
from functools import partial
from itertools import chain

__all__ = ['min_edge_cover', 'is_edge_cover']

[docs]@not_implemented_for('directed')
@not_implemented_for('multigraph')
def min_edge_cover(G, matching_algorithm=None):
"""Returns a set of edges which constitutes
the minimum edge cover of the graph.

A smallest edge cover can be found in polynomial time by finding
a maximum matching and extending it greedily so that all nodes
are covered.

Parameters
----------
G : NetworkX graph
An undirected bipartite graph.

matching_algorithm : function
A function that returns a maximum cardinality matching in a
given bipartite graph. The function must take one input, the
graph G, and return a dictionary mapping each node to its
mate. If not specified,
:func:~networkx.algorithms.bipartite.matching.hopcroft_karp_matching
will be used. Other possibilities include
:func:~networkx.algorithms.bipartite.matching.eppstein_matching,
or matching algorithms in the
:mod:networkx.algorithms.matching module.

Returns
-------
min_cover : set

It contains all the edges of minimum edge cover
in form of tuples. It contains both the edges (u, v) and (v, u)
for given nodes u and v among the edges of minimum edge cover.

Notes
-----
An edge cover of a graph is a set of edges such that every node of
the graph is incident to at least one edge of the set.
The minimum edge cover is an edge covering of smallest cardinality.

Due to its implementation, the worst-case running time of this algorithm
is bounded by the worst-case running time of the function
matching_algorithm.

Minimum edge cover for bipartite graph can also be found using the
function present in :mod:networkx.algorithms.bipartite.covering
"""
if nx.number_of_isolates(G) > 0:
# min_cover does not exist as there is an isolated node
raise nx.NetworkXException(
"Graph has a node with no edge incident on it, "
"so no edge cover exists.")
if matching_algorithm is None:
matching_algorithm = partial(nx.max_weight_matching,
maxcardinality=True)
maximum_matching = matching_algorithm(G)
# min_cover is superset of maximum_matching
try:
min_cover = set(maximum_matching.items())  # bipartite matching case returns dict
except AttributeError:
min_cover = maximum_matching
# iterate for uncovered nodes
uncovered_nodes = set(G) - {v for u, v in min_cover} - {u for u, v in min_cover}
for v in uncovered_nodes:
# Since v is uncovered, each edge incident to v will join it
# with a covered node (otherwise, if there were an edge joining
# uncovered nodes u and v, the maximum matching algorithm
# would have found it), so we can choose an arbitrary edge
# incident to v. (This applies only in a simple graph, not a
# multigraph.)
u = arbitrary_element(G[v])
return min_cover

[docs]@not_implemented_for('directed')
def is_edge_cover(G, cover):
"""Decides whether a set of edges is a valid edge cover of the graph.

Given a set of edges, whether it is an edge covering can
be decided if we just check whether all nodes of the graph
has an edge from the set, incident on it.

Parameters
----------
G : NetworkX graph
An undirected bipartite graph.

cover : set
Set of edges to be checked.

Returns
-------
bool
Whether the set of edges is a valid edge cover of the graph.

Notes
-----
An edge cover of a graph is a set of edges such that every node of
the graph is incident to at least one edge of the set.
"""
return set(G) <= set(chain.from_iterable(cover))