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# -*- coding: utf-8 -*-
"""
**************
Graph Matching
**************
Given a graph G = (V,E), a matching M in G is a set of pairwise non-adjacent
edges; that is, no two edges share a common vertex.
`Wikipedia: Matching <https://en.wikipedia.org/wiki/Matching_(graph_theory)>`_
"""
# Copyright (C) 2011-2012 by
# Nicholas Mancuso <nick.mancuso@gmail.com>
# All rights reserved.
# BSD license.
import networkx as nx
__all__ = ["min_maximal_matching"]
__author__ = """Nicholas Mancuso (nick.mancuso@gmail.com)"""
def min_maximal_matching(G):
r"""Returns the minimum maximal matching of G. That is, out of all maximal
matchings of the graph G, the smallest is returned.
Parameters
----------
G : NetworkX graph
Undirected graph
Returns
-------
min_maximal_matching : set
Returns a set of edges such that no two edges share a common endpoint
and every edge not in the set shares some common endpoint in the set.
Cardinality will be 2*OPT in the worst case.
Notes
-----
The algorithm computes an approximate solution fo the minimum maximal
cardinality matching problem. The solution is no more than 2 * OPT in size.
Runtime is $O(|E|)$.
References
----------
.. [1] Vazirani, Vijay Approximation Algorithms (2001)
"""
return nx.maximal_matching(G)
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