hSrSSKJr SSKJr SSKrSSKJrJ r SS/r \ "S5\ "S 5\RS S j555r \ "S5\RS 55r g) z"Functions related to graph covers.)partial)chainN)arbitrary_elementnot_implemented_formin_edge_cover is_edge_coverdirected multigraphcR[U5S:Xa [5$[R"U5S:a[R"S5eUc[ [R SS9nU"U5n[UR55nSn[U5UVVs1sHupVUiM snn- UVVs1sHupVUiM snn- nUH=n[X5nURXV45 U(dM+URXe45 M? U$![a UnSnNf=fs snnfs snnf)a& Returns the min cardinality edge cover of the graph as a set of edges. A smallest edge cover can be found in polynomial time by finding a maximum matching and extending it greedily so that all nodes are covered. This function follows that process. A maximum matching algorithm can be specified for the first step of the algorithm. The resulting set may return a set with one 2-tuple for each edge, (the usual case) or with both 2-tuples `(u, v)` and `(v, u)` for each edge. The latter is only done when a bipartite matching algorithm is specified as `matching_algorithm`. Parameters ---------- G : NetworkX graph An undirected graph. matching_algorithm : function A function that returns a maximum cardinality matching for `G`. The function must take one input, the graph `G`, and return either a set of edges (with only one direction for the pair of nodes) or a dictionary mapping each node to its mate. If not specified, :func:`~networkx.algorithms.matching.max_weight_matching` is used. Common bipartite matching functions include :func:`~networkx.algorithms.bipartite.matching.hopcroft_karp_matching` or :func:`~networkx.algorithms.bipartite.matching.eppstein_matching`. Returns ------- min_cover : set A set of the edges in a minimum edge cover in the form of tuples. It contains only one of the equivalent 2-tuples `(u, v)` and `(v, u)` for each edge. If a bipartite method is used to compute the matching, the returned set contains both the 2-tuples `(u, v)` and `(v, u)` for each edge of a minimum edge cover. Examples -------- >>> G = nx.Graph([(0, 1), (0, 2), (0, 3), (1, 2), (1, 3)]) >>> sorted(nx.min_edge_cover(G)) [(2, 1), (3, 0)] Notes ----- An edge cover of a graph is a set of edges such that every node of the graph is incident to at least one edge of the set. The minimum edge cover is an edge covering of smallest cardinality. Due to its implementation, the worst-case running time of this algorithm is bounded by the worst-case running time of the function ``matching_algorithm``. Minimum edge cover for `G` can also be found using the `min_edge_covering` function in :mod:`networkx.algorithms.bipartite.covering` which is simply this function with a default matching algorithm of :func:`~networkx.algorithms.bipartite.matching.hopcraft_karp_matching` rzFGraph has a node with no edge incident on it, so no edge cover exists.T)maxcardinalityF) lensetnxnumber_of_isolatesNetworkXExceptionrmax_weight_matchingitemsAttributeErrorradd)Gmatching_algorithmmaximum_matching min_coverbipartite_coveruvuncovered_nodess N/var/www/html/env/lib/python3.13/site-packages/networkx/algorithms/covering.pyrr s| 1v{u  Q!#"" T  !$R%;%;DQ)!, (..01  !fi8idai88);T)$!A);TTO  ad # qf ? MM1& ! !  $   9;Ts.D D0D# DDcX[U5[[R"U55:*$)aDecides whether a set of edges is a valid edge cover of the graph. Given a set of edges, whether it is an edge covering can be decided if we just check whether all nodes of the graph has an edge from the set, incident on it. Parameters ---------- G : NetworkX graph An undirected bipartite graph. cover : set Set of edges to be checked. Returns ------- bool Whether the set of edges is a valid edge cover of the graph. Examples -------- >>> G = nx.Graph([(0, 1), (0, 2), (0, 3), (1, 2), (1, 3)]) >>> cover = {(2, 1), (3, 0)} >>> nx.is_edge_cover(G, cover) True Notes ----- An edge cover of a graph is a set of edges such that every node of the graph is incident to at least one edge of the set. )rr from_iterable)rcovers rrrls$D q6S,,U34 44)N)__doc__ functoolsr itertoolsrnetworkxrnetworkx.utilsrr__all__ _dispatchablerrr"rr+s|(A _ -Z \"Z#!ZzZ  5! 5r"