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"""Provides a function for computing the extendability of a graph which is
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undirected, simple, connected and bipartite and contains at least one perfect matching."""
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import networkx as nx
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from networkx.utils import not_implemented_for
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__all__ = ["maximal_extendability"]
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@not_implemented_for("directed")
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@not_implemented_for("multigraph")
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@nx._dispatchable
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def maximal_extendability(G):
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"""Computes the extendability of a graph.
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The extendability of a graph is defined as the maximum $k$ for which `G`
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is $k$-extendable. Graph `G` is $k$-extendable if and only if `G` has a
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perfect matching and every set of $k$ independent edges can be extended
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to a perfect matching in `G`.
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Parameters
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----------
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G : NetworkX Graph
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A fully-connected bipartite graph without self-loops
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Returns
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-------
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extendability : int
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Raises
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------
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NetworkXError
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If the graph `G` is disconnected.
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If the graph `G` is not bipartite.
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If the graph `G` does not contain a perfect matching.
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If the residual graph of `G` is not strongly connected.
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Notes
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-----
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Definition:
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Let `G` be a simple, connected, undirected and bipartite graph with a perfect
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matching M and bipartition (U,V). The residual graph of `G`, denoted by $G_M$,
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is the graph obtained from G by directing the edges of M from V to U and the
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edges that do not belong to M from U to V.
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Lemma [1]_ :
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Let M be a perfect matching of `G`. `G` is $k$-extendable if and only if its residual
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graph $G_M$ is strongly connected and there are $k$ vertex-disjoint directed
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paths between every vertex of U and every vertex of V.
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Assuming that input graph `G` is undirected, simple, connected, bipartite and contains
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a perfect matching M, this function constructs the residual graph $G_M$ of G and
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returns the minimum value among the maximum vertex-disjoint directed paths between
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every vertex of U and every vertex of V in $G_M$. By combining the definitions
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and the lemma, this value represents the extendability of the graph `G`.
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Time complexity O($n^3$ $m^2$)) where $n$ is the number of vertices
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and $m$ is the number of edges.
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References
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----------
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.. [1] "A polynomial algorithm for the extendability problem in bipartite graphs",
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J. Lakhal, L. Litzler, Information Processing Letters, 1998.
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.. [2] "On n-extendible graphs", M. D. Plummer, Discrete Mathematics, 31:201–210, 1980
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https://doi.org/10.1016/0012-365X(80)90037-0
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"""
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if not nx.is_connected(G):
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raise nx.NetworkXError("Graph G is not connected")
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if not nx.bipartite.is_bipartite(G):
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raise nx.NetworkXError("Graph G is not bipartite")
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U, V = nx.bipartite.sets(G)
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maximum_matching = nx.bipartite.hopcroft_karp_matching(G)
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if not nx.is_perfect_matching(G, maximum_matching):
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raise nx.NetworkXError("Graph G does not contain a perfect matching")
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# list of edges in perfect matching, directed from V to U
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pm = [(node, maximum_matching[node]) for node in V & maximum_matching.keys()]
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# Direct all the edges of G, from V to U if in matching, else from U to V
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directed_edges = [
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(x, y) if (x in V and (x, y) in pm) or (x in U and (y, x) not in pm) else (y, x)
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for x, y in G.edges
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]
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# Construct the residual graph of G
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residual_G = nx.DiGraph()
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residual_G.add_nodes_from(G)
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residual_G.add_edges_from(directed_edges)
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if not nx.is_strongly_connected(residual_G):
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raise nx.NetworkXError("The residual graph of G is not strongly connected")
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# For node-pairs between V & U, keep min of max number of node-disjoint paths
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# Variable $k$ stands for the extendability of graph G
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k = float("inf")
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for u in U:
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for v in V:
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num_paths = sum(1 for _ in nx.node_disjoint_paths(residual_G, u, v))
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k = k if k < num_paths else num_paths
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return k
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