to_vertex_cover¶

to_vertex_cover(G, matching)[source]¶

Returns the minimum vertex cover corresponding to the given maximum matching of the bipartite graph $G$ .

Parameters:

G (NetworkX graph) – Undirected bipartite graph
matching (dictionary) – A dictionary whose keys are vertices in $G$ and whose values are the distinct neighbors comprising the maximum matching for $G$ , as returned by, for example, maximum_matching(). The dictionary must represent the maximum matching.

Returns:

vertex_cover –

The minimum vertex cover in $G$ .

Return type:

set

Notes

This function is implemented using the procedure guaranteed by Konig’s theorem, which proves an equivalence between a maximum matching and a minimum vertex cover in bipartite graphs.

Since a minimum vertex cover is the complement of a maximum independent set for any graph, one can compute the maximum independent set of a bipartite graph this way:

>>> import networkx as nx
>>> G = nx.complete_bipartite_graph(2, 3)
>>> matching = nx.bipartite.maximum_matching(G)
>>> vertex_cover = nx.bipartite.to_vertex_cover(G, matching)
>>> independent_set = set(G) - vertex_cover
>>> print(list(independent_set))
[2, 3, 4]