image-match/python/py/Lib/site-packages/networkx/algorithms/bipartite/extendability.py

"""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