h pSrSSKrSSKJr SSKJr /SQr\"S5\"S5\RS555r \"S5\"S5\RS 555r \"S5\"S5\RS 555r \"S5\"S5\RS 555r g) zCFunctions for computing large cliques and maximum independent sets.N)ramsey)not_implemented_for)clique_removal max_cliquelarge_clique_sizemaximum_independent_setdirected multigraphc [U5upU$)uReturns an approximate maximum independent set. Independent set or stable set is a set of vertices in a graph, no two of which are adjacent. That is, it is a set I of vertices such that for every two vertices in I, there is no edge connecting the two. Equivalently, each edge in the graph has at most one endpoint in I. The size of an independent set is the number of vertices it contains [1]_. A maximum independent set is a largest independent set for a given graph G and its size is denoted $\alpha(G)$. The problem of finding such a set is called the maximum independent set problem and is an NP-hard optimization problem. As such, it is unlikely that there exists an efficient algorithm for finding a maximum independent set of a graph. The Independent Set algorithm is based on [2]_. Parameters ---------- G : NetworkX graph Undirected graph Returns ------- iset : Set The apx-maximum independent set Examples -------- >>> G = nx.path_graph(10) >>> nx.approximation.maximum_independent_set(G) {0, 2, 4, 6, 9} Raises ------ NetworkXNotImplemented If the graph is directed or is a multigraph. Notes ----- Finds the $O(|V|/(log|V|)^2)$ apx of independent set in the worst case. References ---------- .. [1] `Wikipedia: Independent set `_ .. [2] Boppana, R., & Halldórsson, M. M. (1992). Approximating maximum independent sets by excluding subgraphs. BIT Numerical Mathematics, 32(2), 180–196. Springer. )r)Giset_s Z/var/www/html/env/lib/python3.13/site-packages/networkx/algorithms/approximation/clique.pyrrsjQGD KcL[R"U5n[U5up#U$)uFind the Maximum Clique Finds the $O(|V|/(log|V|)^2)$ apx of maximum clique/independent set in the worst case. Parameters ---------- G : NetworkX graph Undirected graph Returns ------- clique : set The apx-maximum clique of the graph Examples -------- >>> G = nx.path_graph(10) >>> nx.approximation.max_clique(G) {8, 9} Raises ------ NetworkXNotImplemented If the graph is directed or is a multigraph. Notes ----- A clique in an undirected graph G = (V, E) is a subset of the vertex set `C \subseteq V` such that for every two vertices in C there exists an edge connecting the two. This is equivalent to saying that the subgraph induced by C is complete (in some cases, the term clique may also refer to the subgraph). A maximum clique is a clique of the largest possible size in a given graph. The clique number `\omega(G)` of a graph G is the number of vertices in a maximum clique in G. The intersection number of G is the smallest number of cliques that together cover all edges of G. https://en.wikipedia.org/wiki/Maximum_clique References ---------- .. [1] Boppana, R., & Halldórsson, M. M. (1992). Approximating maximum independent sets by excluding subgraphs. BIT Numerical Mathematics, 32(2), 180–196. Springer. doi:10.1007/BF01994876 )nx complementr)r cgraphr rs rrrHs$l]]1 FV$GD KrcRUR5n[R"U5up#U/nU/nU(abURU5 [R"U5up#U(aUR U5 U(aUR U5 U(aMb[ U[ S9nXd4$)u=Repeatedly remove cliques from the graph. Results in a $O(|V|/(\log |V|)^2)$ approximation of maximum clique and independent set. Returns the largest independent set found, along with found maximal cliques. Parameters ---------- G : NetworkX graph Undirected graph Returns ------- max_ind_cliques : (set, list) tuple 2-tuple of Maximal Independent Set and list of maximal cliques (sets). Examples -------- >>> G = nx.path_graph(10) >>> nx.approximation.clique_removal(G) ({0, 2, 4, 6, 9}, [{0, 1}, {2, 3}, {4, 5}, {6, 7}, {8, 9}]) Raises ------ NetworkXNotImplemented If the graph is directed or is a multigraph. References ---------- .. [1] Boppana, R., & Halldórsson, M. M. (1992). Approximating maximum independent sets by excluding subgraphs. BIT Numerical Mathematics, 32(2), 180–196. Springer. key)copyr ramsey_R2remove_nodes_fromappendmaxlen)r graphc_ii_icliquesisetsmaxisets rrrsJ FFHE&HCeG EE  $##E*  NN3   LL  %%S!G  rc^^^URmUU4SjmSmUU4SjU5nUH,nXVs1sHnTUT:dMUiM nnT"XST5mM. T$s snf)aFind the size of a large clique in a graph. A *clique* is a subset of nodes in which each pair of nodes is adjacent. This function is a heuristic for finding the size of a large clique in the graph. Parameters ---------- G : NetworkX graph Returns ------- k: integer The size of a large clique in the graph. Examples -------- >>> G = nx.path_graph(10) >>> nx.approximation.large_clique_size(G) 2 Raises ------ NetworkXNotImplemented If the graph is directed or is a multigraph. Notes ----- This implementation is from [1]_. Its worst case time complexity is :math:`O(n d^2)`, where *n* is the number of nodes in the graph and *d* is the maximum degree. This function is a heuristic, which means it may work well in practice, but there is no rigorous mathematical guarantee on the ratio between the returned number and the actual largest clique size in the graph. References ---------- .. [1] Pattabiraman, Bharath, et al. "Fast Algorithms for the Maximum Clique Problem on Massive Graphs with Applications to Overlapping Community Detection." *Internet Mathematics* 11.4-5 (2015): 421--448. See also -------- :func:`networkx.algorithms.approximation.clique.max_clique` A function that returns an approximate maximum clique with a guarantee on the approximation ratio. :mod:`networkx.algorithms.clique` Functions for finding the exact maximum clique in a graph. c>U(d [X25$[UTS9nURU5 XVs1sHnTUU:dMUiM nnT"XU-US-U5$s snf)Nr)rremove) r Usize best_sizeuvN_prime_clique_heuristicdegreess rr.,large_clique_size.._clique_heuristicsey' ' w   d>dgajI&=1d> K9EE?s AArc3>># UHnTUT:dMUv M g7f)N).0r+r*r/s r $large_clique_size..s 51WQZ94QQs  r&)degree)r nodesr+r, neighborsr.r*r/s @@@rrrsmxhhGFI 5 5E  !@1 i(?Q @%aAyA  As AA) __doc__networkxr!networkx.algorithms.approximationrnetworkx.utilsr__all__ _dispatchablerrrrr2rrr?sI4. Z \"3#!3lZ \"5#!5pZ \"/#!/dZ \"H#!Hr