maximal_extendability

maximal_extendability(G)

Computes the extendability of a graph.

The extendability of a graph is defined as the maximum \(k\) for which G is \(k\)-extendable. Graph G is \(k\)-extendable if and only if G has a perfect matching and every set of \(k\) independent edges can be extended to a perfect matching in G.

Parameters:

GNetworkX Graph

A fully-connected bipartite graph without self-loops

Returns:

extendabilityint

Raises:

NetworkXError

If the graph G is disconnected. If the graph G is not bipartite. If the graph G does not contain a perfect matching. If the residual graph of G is not strongly connected.

Notes

Definition: Let G be a simple, connected, undirected and bipartite graph with a perfect matching M and bipartition (U,V). The residual graph of G, denoted by \(G_M\), is the graph obtained from G by directing the edges of M from V to U and the edges that do not belong to M from U to V.

Lemma [1] : Let M be a perfect matching of G. G is \(k\)-extendable if and only if its residual graph \(G_M\) is strongly connected and there are \(k\) vertex-disjoint directed paths between every vertex of U and every vertex of V.

Assuming that input graph G is undirected, simple, connected, bipartite and contains a perfect matching M, this function constructs the residual graph \(G_M\) of G and returns the minimum value among the maximum vertex-disjoint directed paths between every vertex of U and every vertex of V in \(G_M\). By combining the definitions and the lemma, this value represents the extendability of the graph G.

Time complexity O(\(n^3\) \(m^2\))) where \(n\) is the number of vertices and \(m\) is the number of edges.

References

[1]

“A polynomial algorithm for the extendability problem in bipartite graphs”, J. Lakhal, L. Litzler, Information Processing Letters, 1998.

[2]

“On n-extendible graphs”, M. D. Plummer, Discrete Mathematics, 31:201–210, 1980 https://doi.org/10.1016/0012-365X(80)90037-0