hfSrSSKrSSKJr SSKJr SS/r\"S 5\R"S S 9SS j55r \RS 5r g)aFunctions for finding node and edge dominating sets. A `dominating set`_ for an undirected graph *G* with vertex set *V* and edge set *E* is a subset *D* of *V* such that every vertex not in *D* is adjacent to at least one member of *D*. An `edge dominating set`_ is a subset *F* of *E* such that every edge not in *F* is incident to an endpoint of at least one edge in *F*. .. _dominating set: https://en.wikipedia.org/wiki/Dominating_set .. _edge dominating set: https://en.wikipedia.org/wiki/Edge_dominating_set N)not_implemented_for)maximal_matchingmin_weighted_dominating_setmin_edge_dominating_setdirectedweight) node_attrsc H^^^[T5S:Xa [5$[5mUUU4Sjn[T5nTVs0sHoDU1[TU5-_M nnU(a:[UR5US9upgTR U5 XV X7-nU(aM:T$s snf)aReturns a dominating set that approximates the minimum weight node dominating set. Parameters ---------- G : NetworkX graph Undirected graph. weight : string The node attribute storing the weight of an node. If provided, the node attribute with this key must be a number for each node. If not provided, each node is assumed to have weight one. Returns ------- min_weight_dominating_set : set A set of nodes, the sum of whose weights is no more than `(\log w(V)) w(V^*)`, where `w(V)` denotes the sum of the weights of each node in the graph and `w(V^*)` denotes the sum of the weights of each node in the minimum weight dominating set. Examples -------- >>> G = nx.Graph([(0, 1), (0, 4), (1, 4), (1, 2), (2, 3), (3, 4), (2, 5)]) >>> nx.approximation.min_weighted_dominating_set(G) {1, 2, 4} Raises ------ NetworkXNotImplemented If G is directed. Notes ----- This algorithm computes an approximate minimum weighted dominating set for the graph `G`. The returned solution has weight `(\log w(V)) w(V^*)`, where `w(V)` denotes the sum of the weights of each node in the graph and `w(V^*)` denotes the sum of the weights of each node in the minimum weight dominating set for the graph. This implementation of the algorithm runs in $O(m)$ time, where $m$ is the number of edges in the graph. References ---------- .. [1] Vazirani, Vijay V. *Approximation Algorithms*. Springer Science & Business Media, 2001. rch>UupTRURTS5[UT- 5- $)zReturns the cost-effectiveness of greedily choosing the given node. `node_and_neighborhood` is a two-tuple comprising a node and its closed neighborhood. )nodesgetlen)node_and_neighborhoodv neighborhoodGdom_setr s b/var/www/html/env/lib/python3.13/site-packages/networkx/algorithms/approximation/dominating_set.py_cost*min_weighted_dominating_set.._costSs40wwqz~~fa(3|g/E+FFF)key)rsetminitemsadd) rr rverticesr neighborhoodsdom_nodemin_setrs `` @rrrsl 1v{u eG G1vH233Ac!A$i'M3  3 3 55A  H  # ( N4sBc<U(d [S5e[U5$)arReturns minimum cardinality edge dominating set. Parameters ---------- G : NetworkX graph Undirected graph Returns ------- min_edge_dominating_set : set Returns a set of dominating edges whose size is no more than 2 * OPT. Examples -------- >>> G = nx.petersen_graph() >>> nx.approximation.min_edge_dominating_set(G) {(0, 1), (4, 9), (6, 8), (5, 7), (2, 3)} Raises ------ ValueError If the input graph `G` is empty. Notes ----- The algorithm computes an approximate solution to the edge dominating set problem. The result is no more than 2 * OPT in terms of size of the set. Runtime of the algorithm is $O(|E|)$. z"Expected non-empty NetworkX graph!) ValueErrorr)rs rrrts> =>> A r)N) __doc__networkxnxutilsrmatchingr__all__ _dispatchablerrrrr.si (' (*C DZ X&X'!Xv  r