wgrdZddlZddlmZdgZededej dZy)zProvides a function for computing the extendability of a graph which is undirected, simple, connected and bipartite and contains at least one perfect matching.N)not_implemented_formaximal_extendabilitydirected multigraphc tj|stjdtjj |stjdtjj |\}}tjj |}tj||stjd||jzDcgc] }|||f }}|jDcgc]!\}}||vr||f|vs ||vr ||f|vr||fn||f#}}}tj} | j|| j|tj| stjdtd} |D]9} |D]2} tdtj | | | D} | | kr| n| } 4;| Scc}wcc}}w)uxComputes 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 zGraph G is not connectedzGraph G is not bipartitez+Graph G does not contain a perfect matchingz1The residual graph of G is not strongly connectedinfc3 K|]}dyw)N).0_s p/home/mcse/projects/flask/flask-venv/lib/python3.12/site-packages/networkx/algorithms/bipartite/extendability.py z(maximal_extendability..gsP!APs )nx is_connected NetworkXError bipartite is_bipartitesetshopcroft_karp_matchingis_perfect_matchingkeysedgesDiGraphadd_nodes_fromadd_edges_fromis_strongly_connectedfloatsumnode_disjoint_paths)GUVmaximum_matchingnodepmxydirected_edges residual_Gkuv num_pathss rrr st ??1 9:: << $ $Q '9:: <<  Q DAq||::1= ! !!%5 6LMM679I9N9N9P5P QT4!$' ( QB Q GG Aq6q!flQAq6;KASTVWRXXN Ja n- # #J /RSS e A 2 2APr'='=j!Q'OPPI] A 22 H/ Rs G?&G)__doc__networkxrnetworkx.utilsr__all__ _dispatchablerr rr5sQ[. " #Z \"\ #!\ r4