a ‘ Àh…ã@s:dZddlZddlmZdgZedƒedƒdd„ƒƒZdS)z  Provides a function for computing the extendability of a graph which is undirected, simple, connected and bipartite and contains at least one perfect matching.éN)Únot_implemented_forÚmaximal_extendabilityZdirectedZ multigraphc s t |¡st d¡‚tj |¡s*t d¡‚tj |¡\‰‰tj |¡‰t |ˆ¡s\t d¡‚‡fdd„ˆˆ ¡@Dƒ‰‡‡‡fdd„|j Dƒ}t  ¡}|  |¡|  |¡t  |¡s¾t d¡‚tdƒ}ˆD]:}ˆD]0}td d „t |||¡Dƒƒ}||krþ|n|}qÒqÊ|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 matchingcsg|]}|ˆ|f‘qS©r)Ú.0Únode)Úmaximum_matchingrúg/var/www/html/assistant/venv/lib/python3.9/site-packages/networkx/algorithms/bipartite/extendability.pyÚ Róz)maximal_extendability..csHg|]@\}}|ˆvr ||fˆvs4|ˆvr<||fˆvr<||fn||f‘qSrr)rÚxÚy)ÚUÚVÚpmrrr Usÿz1The residual graph of G is not strongly connectedZInfcss|] }dVqdS)éNr)rÚ_rrrÚ gr z(maximal_extendability..)ÚnxZ is_connectedZ NetworkXErrorZ bipartiteZ is_bipartiteZsetsZhopcroft_karp_matchingZis_perfect_matchingÚkeysÚedgesZDiGraphZadd_nodes_fromZadd_edges_fromZis_strongly_connectedÚfloatÚsumZnode_disjoint_paths)ÚGZdirected_edgesZ residual_GÚkÚuÚvZ num_pathsr)r rrrrr s.9       þ    )Ú__doc__ZnetworkxrZnetworkx.utilsrÚ__all__rrrrrÚs