+ )i%Rt^RIHtHt^RIt^RIHt.ROt]!R4]!R4]PR444t Rt ]PR4t ]PR 4t ]PR 4t]!R4]!R4]P!R R 7RR l444t]!R4]!R4]P!R R 7RRl444tR#)z;Functions for computing and verifying matchings in a graph.) combinationsrepeatN)not_implemented_for multigraphdirectedc\4p\4pVP4FApVwrEWB9gKWR9gKWE8wgKVPV4VPV4KC V#)aIFind a maximal matching in the graph. A matching is a subset of edges in which no node occurs more than once. A maximal matching cannot add more edges and still be a matching. Parameters ---------- G : NetworkX graph Undirected graph Returns ------- matching : set A maximal matching of the graph. Examples -------- >>> G = nx.Graph([(1, 2), (1, 3), (2, 3), (2, 4), (3, 5), (4, 5)]) >>> sorted(nx.maximal_matching(G)) [(1, 2), (3, 5)] Notes ----- The algorithm greedily selects a maximal matching M of the graph G (i.e. no superset of M exists). It runs in $O(|E|)$ time. )setedgesaddupdateGmatchingnodesedgeuvs& Z/var/www/html/photoedit/myenv/lib/python3.14/site-packages/networkx/algorithms/matching.pymaximal_matchingrsW<uH EE  >an LL  LL   Oc\4pVP4FGpVwr4WC3V9gW!9dKW48Xd\P!RV 24hVP V4KI V#)aConverts matching dict format to matching set format Converts a dictionary representing a matching (as returned by :func:`max_weight_matching`) to a set representing a matching (as returned by :func:`maximal_matching`). In the definition of maximal matching adopted by NetworkX, self-loops are not allowed, so the provided dictionary is expected to never have any mapping from a key to itself. However, the dictionary is expected to have mirrored key/value pairs, for example, key ``u`` with value ``v`` and key ``v`` with value ``u``. z%Selfloops cannot appear in matchings )ritemsnx NetworkXErrorr )rr rrrs& rmatching_dict_to_setr<sc EE  6U?dm  6""%J4&#QR R $ ! Lrc\V\4'd \V4p\4pVFp\ V4^8wd\ P !RV 24hVwrEW@9gWP9d\ P !RV R24hWE8XdR#VPWE4'gR#WB9gWR9dR#VPV4K R#)a[Return True if ``matching`` is a valid matching of ``G`` A *matching* in a graph is a set of edges in which no two distinct edges share a common endpoint. Each node is incident to at most one edge in the matching. The edges are said to be independent. Parameters ---------- G : NetworkX graph matching : dict or set A dictionary or set representing a matching. If a dictionary, it must have ``matching[u] == v`` and ``matching[v] == u`` for each edge ``(u, v)`` in the matching. If a set, it must have elements of the form ``(u, v)``, where ``(u, v)`` is an edge in the matching. Returns ------- bool Whether the given set or dictionary represents a valid matching in the graph. Raises ------ NetworkXError If the proposed matching has an edge to a node not in G. Or if the matching is not a collection of 2-tuple edges. Examples -------- >>> G = nx.Graph([(1, 2), (1, 3), (2, 3), (2, 4), (3, 5), (4, 5)]) >>> nx.is_maximal_matching(G, {1: 3, 2: 4}) # using dict to represent matching True >>> nx.is_matching(G, {(1, 3), (2, 4)}) # using set to represent matching True matching has non-2-tuple edge matching contains edge  with node not in GFT isinstancedictrrlenrrhas_edger r s&& r is_matchingr$UsR(D!!'1 EE t9>""%CD6#JK K :""%>> G = nx.Graph([(1, 2), (1, 3), (2, 3), (3, 4), (3, 5)]) >>> nx.is_maximal_matching(G, {(1, 2), (3, 4)}) True rrrFT) r r!rrr"rrr#r r r )r rr rrrrs&& ris_maximal_matchingr&s>(D!!'1 EE EE t9>""%CD6#JK K :""%>> G = nx.Graph([(1, 2), (1, 3), (2, 3), (2, 4), (3, 5), (4, 5), (4, 6)]) >>> my_match = {1: 2, 3: 5, 4: 6} >>> nx.is_perfect_matching(G, my_match) True rrrFrr s&& ris_perfect_matchingr(s@(D!!'1 EE t9>""%CD6#JK K :""%&min_weight_matching..7s2'wq!'sc3@<"TFwrq1VSV, 3xK R#5ir0r1)r2rrr4 max_weights& rr5r69s ;7aJN #7sr))r"r max_weight_matchingmaxrGraphadd_weighted_edges_from)r r)G_edgesInvGr r8s&& @rmin_weight_matchingr@s` 177|q"1T&IIgg61g-GS2'222J 88:D ;7 ;E   6 tD HHrc aaaaaaaaaa a!a"a#a$a%a&a'a(a)a*!RR4o!V3RlR4o\S4o$S$'g \4#^pRpSPRR7F]wrVpVPS^4pWV8wd W8dTpT;'d.\ \ V44P R4^,R9pK_ /o(/o&/o'\\S$S$44o%\\S$\R444o"\\S$S$44o /o\\S$\V444o#/o!/o.o)VV#V3R lo*VVVV V%V&V'V(V)3 R loVV V%V&V'V(3R lp VVVV V!V"V%V&V'V(V)V*3 R lp VVVVV V!V"V%V&V'V(3 R lp VV V"V(3RloVVV V%V&V'V(3Rlp VV!V"V#V$V(VV3Rlp S&P4S'P4SP4S!F pRVn K SP4.S)R&S$F2pVS(9gK S&PS%V,4eK(S!V^R4K4 ^pS)'EdV'EgS)P4pS&S%V,,^8XgQhSPV4EFwpVV8XdK S%V,pS%V,pVV8XdK(VV3S9dS*!VV4pV^8:dR;SVV3&SVV3&VV3S9dS&PV4f S!V^V4KwS&PV4^8Xd+V !VV4pVSJd V !VVV4KV !VV4^pEKS&PV4fS&V,^8XgQh^S&V&VV3S'V&KKS&PV4^8Xd3SPV4eXS*!SV,!8d VV3SV&EK/EK2S&PV4eEKHSPV4eXS*!SV,!8gEKpVV3SV&EKz EKV'dEMFRpR;p;ppS'g^p\S#P44pSP!4F^pS&PS%V,4eKSPV4fK4S*!SV,!pVR8Xg VV8gKQTp^pSV,pK` S"FpS"V,eKS&PV4^8XgK(SPV4fK=S*!SV,!pV'dV^,^8XgQhV^,pM VR, pVR8Xg VV8gKTp^pSV,pK S!FLpS"V,eKS&PV4^8XgK(VR8XgS!V,V8gK?S!V,p^pTpKN VR8Xd0S'gQh^p\#^\S#P444pS$FkpS&PS%V,4^8XdS#V;;,V,uu&K7S&PS%V,4^8XgKVS#V;;,V, uu&Km S!FjpS"V,eKS&PV4^8XdS!V;;,V, uu&K=S&PV4^8XgKUS!V;;,V,uu&Kl V^8XdMV^8Xd?VwppS&S%V,,^8XgQhR;SVV3&SVV3&S)P%V4EKV^8Xd?VwppR;SVV3&SVV3&S&S%V,,^8XgQhS)P%V4EKV^8XgEKV !VR4EKS(FpS(S(V,,V8XdKQh V'gMj\S!P'44FJpVS!9dK S"V,eKS&PV4^8XgK1S!V,^8XgKAV !VR4KL EK=V'dV !4\)S(4#)aCompute a maximum-weighted matching of G. A matching is a subset of edges in which no node occurs more than once. The weight of a matching is the sum of the weights of its edges. A maximal matching cannot add more edges and still be a matching. The cardinality of a matching is the number of matched edges. Parameters ---------- G : NetworkX graph Undirected graph maxcardinality: bool, optional (default=False) If maxcardinality is True, compute the maximum-cardinality matching with maximum weight among all maximum-cardinality matchings. weight: string, optional (default='weight') Edge data key corresponding to the edge weight. If key not found, uses 1 as weight. Returns ------- matching : set A maximal matching of the graph. Examples -------- >>> G = nx.Graph() >>> edges = [(1, 2, 6), (1, 3, 2), (2, 3, 1), (2, 4, 7), (3, 5, 9), (4, 5, 3)] >>> G.add_weighted_edges_from(edges) >>> sorted(nx.max_weight_matching(G)) [(2, 4), (5, 3)] Notes ----- If G has edges with weight attributes the edge data are used as weight values else the weights are assumed to be 1. This function takes time O(number_of_nodes ** 3). If all edge weights are integers, the algorithm uses only integer computations. If floating point weights are used, the algorithm could return a slightly suboptimal matching due to numeric precision errors. This method is based on the "blossom" method for finding augmenting paths and the "primal-dual" method for finding a matching of maximum weight, both methods invented by Jack Edmonds [1]_. Bipartite graphs can also be matched using the functions present in :mod:`networkx.algorithms.bipartite.matching`. References ---------- .. [1] "Efficient Algorithms for Finding Maximum Matching in Graphs", Zvi Galil, ACM Computing Surveys, 1986. c]tRtRtRtRtR#)#max_weight_matching..NoNodeiz-Dummy value which is different from any node.r1N)__name__ __module__ __qualname____firstlineno____doc____static_attributes__r1rrNoNoderCs;rrJc:<a]tRtRtoRt.ROtV3RltRtVtR#)$max_weight_matching..Blossomiz7Representation of a non-trivial blossom or sub-blossom.c3<".VPOpV'dFVP4p\VS4'dVPVP4KGVxKMR#5ir0)childspopr extend)selfstacktBlossoms& rleaves+max_weight_matching..Blossom.leavessH"dkkNEIIKa))LL*G s A!AA!r1N)rNr mybestedges) rDrErFrGrH __slots__rUrI__classdictcell__) __classdict__rTs@rrTrLsE6   rrTTr-'Nc<SV,SV,,^SV,V,PS^4,, #))get)rr4r dualvarr)s&&rslack"max_weight_matching..slacks3qzGAJ&QqT!W[[-C)CCCrc< S V,pS PV4fS PV4eQhV;S V&S V&Ve W 3;S V&S V&M R;S V&S V&R;SV&SV&V^8XdG\VS4'd"S PVP44R#S P V4R#V^8XdSV,pS!S V,^V4R#R#r0)r_r rPrUappend)r4rSrbbaserT assignLabelbestedge blossombase inblossomlabel labeledgematequeues&&& rrg(max_weight_matching..assignLabels aLyy|# ! 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