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+"""Provides a function for computing the extendability of a graph which is
+undirected, simple, connected and bipartite and contains at least one perfect matching."""
+
+import networkx as nx
+from networkx.utils import not_implemented_for
+
+__all__ = ["maximal_extendability"]
+
+
+@not_implemented_for("directed")
+@not_implemented_for("multigraph")
+@nx._dispatchable
+def 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
+ ----------
+ G : NetworkX Graph
+ A fully-connected bipartite graph without self-loops
+
+ Returns
+ -------
+ extendability : int
+
+ 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
+
+ """
+ if not nx.is_connected(G):
+ raise nx.NetworkXError("Graph G is not connected")
+
+ if not nx.bipartite.is_bipartite(G):
+ raise nx.NetworkXError("Graph G is not bipartite")
+
+ U, V = nx.bipartite.sets(G)
+
+ maximum_matching = nx.bipartite.hopcroft_karp_matching(G)
+
+ if not nx.is_perfect_matching(G, maximum_matching):
+ raise nx.NetworkXError("Graph G does not contain a perfect matching")
+
+ # list of edges in perfect matching, directed from V to U
+ pm = [(node, maximum_matching[node]) for node in V & maximum_matching.keys()]
+
+ # Direct all the edges of G, from V to U if in matching, else from U to V
+ directed_edges = [
+ (x, y) if (x in V and (x, y) in pm) or (x in U and (y, x) not in pm) else (y, x)
+ for x, y in G.edges
+ ]
+
+ # Construct the residual graph of G
+ residual_G = nx.DiGraph()
+ residual_G.add_nodes_from(G)
+ residual_G.add_edges_from(directed_edges)
+
+ if not nx.is_strongly_connected(residual_G):
+ raise nx.NetworkXError("The residual graph of G is not strongly connected")
+
+ # For node-pairs between V & U, keep min of max number of node-disjoint paths
+ # Variable $k$ stands for the extendability of graph G
+ k = float("inf")
+ for u in U:
+ for v in V:
+ num_paths = sum(1 for _ in nx.node_disjoint_paths(residual_G, u, v))
+ k = k if k < num_paths else num_paths
+ return k