btjdZddlZddlmZdgZededejdZdS)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 tj|stjdtj|stjdtj|\tj| tj| stjd fd zD  fd|j D}tj }| || |tj |stjdtd}D]>}D]9}tdtj|||D}||kr|n|}:?|S) 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 matchingc$g|] }||f Sr ).0nodemaximum_matchings i/var/www/html/CA-Chatbot/venv/lib/python3.11/site-packages/networkx/algorithms/bipartite/extendability.py z)maximal_extendability..Rs$ Q Q QT4!$' ( Q Q QcNg|]!\}}|vr||fvs |vr ||fvr||fn||f"Sr r )r xyUVpms r rz)maximal_extendability..Usf Aq66q!fllQAq6;K;KASTVWRXrz1The residual graph of G is not strongly connectedinfc3K|]}dVdS)Nr )r _s r z(maximal_extendability..gs"PP!APPPPPPr)nx is_connected NetworkXError bipartite is_bipartitesetshopcroft_karp_matchingis_perfect_matchingkeysedgesDiGraphadd_nodes_fromadd_edges_fromis_strongly_connectedfloatsumnode_disjoint_paths) Gdirected_edges residual_Gkuv num_pathsrrr rs @@@@r rr st ?1  ;9::: < $ $Q ' ';9::: <  Q  DAq|::1== !!%5 6 6NLMMM R Q Q QQ9I9N9N9P9P5P Q Q QBGN Ja   n--- #J / /TRSSS e A 22 2 2APPr'=j!Q'O'OPPPPPI]] AA 2 Hr)__doc__networkxrnetworkx.utilsr__all__ _dispatchablerr rr r8s[[...... " #Z  \""\ \ #"! \ \ \ r