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# Source code for networkx.algorithms.bipartite.extendability

    """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