~irSrSSKrSSKJr S/r\"S5\"S5\R S555rg)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 [R"U5(d[R"S5e[RR U5(d[R"S5e[RR U5up[RR U5n[R"X5(d[R"S5eX#R5-Vs/sHoDX44PM nnURVVs/sH#upgXb;aXg4U;d Xa;a Xv4U;aXg4OXv4PM% nnn[R"5n U RU5 U RU5 [R"U 5(d[R"S5e[S5n UH<n UH3n [S[R "XU 555n X:aU OU n M5 M> U $s snfs snnf)uComputes 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&# UHnSv M g7f)N).0_s h/home/mande/repo/quber/.venv/lib/python3.13/site-packages/networkx/algorithms/bipartite/extendability.py (maximal_extendability..gsP'O!A'Os)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 DA||::1= ! !! 6 6LMM679N9N9P5P Q5PT!' (5PB Q GGDA6qflA6;KSTRXX Ja n- # #J / /RSS e A APr'='=jQ'OPPI] A H/ Rs 1G5*G:)__doc__networkxrnetworkx.utilsr__all__ _dispatchablerr rr6sQ[. " #Z \"\ #!\ r5