# Copyright 2016-2019 NetworkX developers.
# Copyright (C) 2016 by
# Nishant Nikhil <nishantiam@gmail.com>
# All rights reserved.
# BSD license.
""" 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])
min_cover.add((u, v))
min_cover.add((v, u))
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))