hQVSrSSKrSSKJr SSKrSSKJrJ r /SQr \ "S5\RS55r \ "S5\RS55r S r\ "S 5\ "S5\RS 555r"S S 5rSSjrSr\R"SS9S5rg)aw Algorithms for finding k-edge-connected components and subgraphs. A k-edge-connected component (k-edge-cc) is a maximal set of nodes in G, such that all pairs of node have an edge-connectivity of at least k. A k-edge-connected subgraph (k-edge-subgraph) is a maximal set of nodes in G, such that the subgraph of G defined by the nodes has an edge-connectivity at least k. N)partial)arbitrary_elementnot_implemented_for)k_edge_componentsk_edge_subgraphsbridge_componentsEdgeComponentAuxGraph multigraphcxUS:a [S5eUR5(aBUS:Xa[R"U5$[R U5nUR U5$US:Xa[R"U5$US:Xa [U5$[R U5nUR U5$)aGenerates nodes in each maximal k-edge-connected component in G. Parameters ---------- G : NetworkX graph k : Integer Desired edge connectivity Returns ------- k_edge_components : a generator of k-edge-ccs. Each set of returned nodes will have k-edge-connectivity in the graph G. See Also -------- :func:`local_edge_connectivity` :func:`k_edge_subgraphs` : similar to this function, but the subgraph defined by the nodes must also have k-edge-connectivity. :func:`k_components` : similar to this function, but uses node-connectivity instead of edge-connectivity Raises ------ NetworkXNotImplemented If the input graph is a multigraph. ValueError: If k is less than 1 Notes ----- Attempts to use the most efficient implementation available based on k. If k=1, this is simply connected components for directed graphs and connected components for undirected graphs. If k=2 on an efficient bridge connected component algorithm from _[1] is run based on the chain decomposition. Otherwise, the algorithm from _[2] is used. Examples -------- >>> import itertools as it >>> from networkx.utils import pairwise >>> paths = [ ... (1, 2, 4, 3, 1, 4), ... (5, 6, 7, 8, 5, 7, 8, 6), ... ] >>> G = nx.Graph() >>> G.add_nodes_from(it.chain(*paths)) >>> G.add_edges_from(it.chain(*[pairwise(path) for path in paths])) >>> # note this returns {1, 4} unlike k_edge_subgraphs >>> sorted(map(sorted, nx.k_edge_components(G, k=3))) [[1, 4], [2], [3], [5, 6, 7, 8]] References ---------- .. [1] https://en.wikipedia.org/wiki/Bridge_%28graph_theory%29 .. [2] Wang, Tianhao, et al. (2015) A simple algorithm for finding all k-edge-connected components. http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0136264 k cannot be less than 1) ValueError is_directednxstrongly_connected_componentsr constructrconnected_componentsr)Gk aux_graphs c/var/www/html/env/lib/python3.13/site-packages/networkx/algorithms/connectivity/edge_kcomponents.pyrrsB 1u233}} 633A6 6.77:I..q1 1 6**1- - !V$Q' '-77:I..q1 1cUS:a [S5eUR5(aUS::a [X5$[X5$US::a [X5$[X5$)u=Generates nodes in each maximal k-edge-connected subgraph in G. Parameters ---------- G : NetworkX graph k : Integer Desired edge connectivity Returns ------- k_edge_subgraphs : a generator of k-edge-subgraphs Each k-edge-subgraph is a maximal set of nodes that defines a subgraph of G that is k-edge-connected. See Also -------- :func:`edge_connectivity` :func:`k_edge_components` : similar to this function, but nodes only need to have k-edge-connectivity within the graph G and the subgraphs might not be k-edge-connected. Raises ------ NetworkXNotImplemented If the input graph is a multigraph. ValueError: If k is less than 1 Notes ----- Attempts to use the most efficient implementation available based on k. If k=1, or k=2 and the graph is undirected, then this simply calls `k_edge_components`. Otherwise the algorithm from _[1] is used. Examples -------- >>> import itertools as it >>> from networkx.utils import pairwise >>> paths = [ ... (1, 2, 4, 3, 1, 4), ... (5, 6, 7, 8, 5, 7, 8, 6), ... ] >>> G = nx.Graph() >>> G.add_nodes_from(it.chain(*paths)) >>> G.add_edges_from(it.chain(*[pairwise(path) for path in paths])) >>> # note this does not return {1, 4} unlike k_edge_components >>> sorted(map(sorted, nx.k_edge_subgraphs(G, k=3))) [[1], [2], [3], [4], [5, 6, 7, 8]] References ---------- .. [1] Zhou, Liu, et al. (2012) Finding maximal k-edge-connected subgraphs from a large graph. ACM International Conference on Extending Database Technology 2012 480-–491. https://openproceedings.org/2012/conf/edbt/ZhouLYLCL12.pdf r r r)rrr_k_edge_subgraphs_nodes)rrs rrrnsaz 1u233}} 6%Q* **10 0 6%Q* **10 0rc#f# [X5Hn[UR55v M g7f)zaHelper to get the nodes from the subgraphs. This allows k_edge_subgraphs to return a generator. N)general_k_edge_subgraphssetnodes)rrCs rrrs& &a +!'')n,s/1directedc## UR5nUR[R"U55 [R"U5ShvN gN7f)auFinds all bridge-connected components G. Parameters ---------- G : NetworkX undirected graph Returns ------- bridge_components : a generator of 2-edge-connected components See Also -------- :func:`k_edge_subgraphs` : this function is a special case for an undirected graph where k=2. :func:`biconnected_components` : similar to this function, but is defined using 2-node-connectivity instead of 2-edge-connectivity. Raises ------ NetworkXNotImplemented If the input graph is directed or a multigraph. Notes ----- Bridge-connected components are also known as 2-edge-connected components. Examples -------- >>> # The barbell graph with parameter zero has a single bridge >>> G = nx.barbell_graph(5, 0) >>> from networkx.algorithms.connectivity.edge_kcomponents import bridge_components >>> sorted(map(sorted, bridge_components(G))) [[0, 1, 2, 3, 4], [5, 6, 7, 8, 9]] N)copyremove_edges_fromrbridgesr)rHs rrrs=N A 1 &&&q)))sAAAAc4\rSrSrSr\S5rSrSrSr g)r a A simple algorithm to find all k-edge-connected components in a graph. Constructing the auxiliary graph (which may take some time) allows for the k-edge-ccs to be found in linear time for arbitrary k. Notes ----- This implementation is based on [1]_. The idea is to construct an auxiliary graph from which the k-edge-ccs can be extracted in linear time. The auxiliary graph is constructed in $O(|V|\cdot F)$ operations, where F is the complexity of max flow. Querying the components takes an additional $O(|V|)$ operations. This algorithm can be slow for large graphs, but it handles an arbitrary k and works for both directed and undirected inputs. The undirected case for k=1 is exactly connected components. The undirected case for k=2 is exactly bridge connected components. The directed case for k=1 is exactly strongly connected components. References ---------- .. [1] Wang, Tianhao, et al. (2015) A simple algorithm for finding all k-edge-connected components. http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0136264 Examples -------- >>> import itertools as it >>> from networkx.utils import pairwise >>> from networkx.algorithms.connectivity import EdgeComponentAuxGraph >>> # Build an interesting graph with multiple levels of k-edge-ccs >>> paths = [ ... (1, 2, 3, 4, 1, 3, 4, 2), # a 3-edge-cc (a 4 clique) ... (5, 6, 7, 5), # a 2-edge-cc (a 3 clique) ... (1, 5), # combine first two ccs into a 1-edge-cc ... (0,), # add an additional disconnected 1-edge-cc ... ] >>> G = nx.Graph() >>> G.add_nodes_from(it.chain(*paths)) >>> G.add_edges_from(it.chain(*[pairwise(path) for path in paths])) >>> # Constructing the AuxGraph takes about O(n ** 4) >>> aux_graph = EdgeComponentAuxGraph.construct(G) >>> # Once constructed, querying takes O(n) >>> sorted(map(sorted, aux_graph.k_edge_components(k=1))) [[0], [1, 2, 3, 4, 5, 6, 7]] >>> sorted(map(sorted, aux_graph.k_edge_components(k=2))) [[0], [1, 2, 3, 4], [5, 6, 7]] >>> sorted(map(sorted, aux_graph.k_edge_components(k=3))) [[0], [1, 2, 3, 4], [5], [6], [7]] >>> sorted(map(sorted, aux_graph.k_edge_components(k=4))) [[0], [1], [2], [3], [4], [5], [6], [7]] The auxiliary graph is primarily used for k-edge-ccs but it can also speed up the queries of k-edge-subgraphs by refining the search space. >>> import itertools as it >>> from networkx.utils import pairwise >>> from networkx.algorithms.connectivity import EdgeComponentAuxGraph >>> paths = [ ... (1, 2, 4, 3, 1, 4), ... ] >>> G = nx.Graph() >>> G.add_nodes_from(it.chain(*paths)) >>> G.add_edges_from(it.chain(*[pairwise(path) for path in paths])) >>> aux_graph = EdgeComponentAuxGraph.construct(G) >>> sorted(map(sorted, aux_graph.k_edge_subgraphs(k=3))) [[1], [2], [3], [4]] >>> sorted(map(sorted, aux_graph.k_edge_components(k=3))) [[1, 4], [2], [3]] c^[S5"S5"U5 U4SjmUR5nURUR55 UR UR 5SS9 [ R"5nUR5S:a;[UR55n[UR55nT"X#XE5 U"5nX6l X&l U$)aBuilds an auxiliary graph encoding edge-connectivity between nodes. Notes ----- Given G=(V, E), initialize an empty auxiliary graph A. Choose an arbitrary source node s. Initialize a set N of available nodes (that can be used as the sink). The algorithm picks an arbitrary node t from N - {s}, and then computes the minimum st-cut (S, T) with value w. If G is directed the minimum of the st-cut or the ts-cut is used instead. Then, the edge (s, t) is added to the auxiliary graph with weight w. The algorithm is called recursively first using S as the available nodes and s as the source, and then using T and t. Recursion stops when the source is the only available node. Parameters ---------- G : NetworkX graph r cU$N)rs r1EdgeComponentAuxGraph.construct..QsArc^>U1U:Xag[X21- 5n[R"XU5unupgUR5(a%[R"XU5unupX:aXU pvnUR X$US9 T "XX#R U55 T "XXCR U55 g)N)weight)rr minimum_cutradd_edge intersection) r&AsourceavailsinkvalueSTvalue_T_S__recursive_builds rr>9EdgeComponentAuxGraph.construct.._recursive_buildSsx5 $UX%56DNN1d;ME6A}}#%>>!6#B >"(baE JJvEJ 2 Q6+=+=a+@ A Q4););A)> ?rr )capacityr) r __class__add_nodes_fromradd_edges_fromedgesrGraphnumber_of_nodesrrr4r&)r rr&r4r5r6selfr>s @rrEdgeComponentAuxGraph.construct;s, L)+6q9 @( KKM # Q/ HHJ    "&qwwy1F NE Q6 1%& rc#p^# TS:a [S5eURn[R"US5n[R"5nUR UR 55 URU4SjUR555 [R"U5ShvN gN7f)aQueries the auxiliary graph for k-edge-connected components. Parameters ---------- k : Integer Desired edge connectivity Returns ------- k_edge_components : a generator of k-edge-ccs Notes ----- Given the auxiliary graph, the k-edge-connected components can be determined in linear time by removing all edges with weights less than k from the auxiliary graph. The resulting connected components are the k-edge-ccs in the original graph. r r r0c3<># UHupUT:dM Uv M g7fr+r,.0ewrs r :EdgeComponentAuxGraph.k_edge_components..E':tqa1f':  N) rr4rget_edge_attributesrErBrrCitemsr)rGrr4 aux_weightsRs ` rr'EdgeComponentAuxGraph.k_edge_componentss& q567 7 FF,,Q9 HHJ # E{'8'8':EE**1---sB+B6.B4/B6c#^# TS:a [S5eURnURn[R"US5n[R "5nUR UR55 URU4SjUR555 [R"U5HGn[U5T:aUHnU1v M M"URU5n[UT5ShvN MI gN 7f)aQueries the auxiliary graph for k-edge-connected subgraphs. Parameters ---------- k : Integer Desired edge connectivity Returns ------- k_edge_subgraphs : a generator of k-edge-subgraphs Notes ----- Refines the k-edge-ccs into k-edge-subgraphs. The running time is more than $O(|V|)$. For single values of k it is faster to use `nx.k_edge_subgraphs`. But for multiple values of k, it can be faster to build AuxGraph and then use this method. r r r0c3<># UHupUT:dM Uv M g7fr+r,rKs rrO9EdgeComponentAuxGraph.k_edge_subgraphs..rQrRN)rr&r4rrSrErBrrCrTrlensubgraphr) rGrr&r4rUrVccnoder s ` rr&EdgeComponentAuxGraph.k_edge_subgraphss* q567 7 FF FF,,Q9 HHJ # E{'8'8':EE))!,B2w{D&LJJrN+Aq111-2sC6D9D: D)r4r&N) __name__ __module__ __qualname____firstlineno____doc__ classmethodrrr__static_attributes__r,rrr r s+EPAAF.B*2rr c#|# UR5(a~[5nURU5H!upEXQ:dM URU5 Uv M# UR U5H(upEXC;dM XQ:dMURU5 Uv M* gUR U5HupEXQ:dM Uv M g7f)z1Helper for finding nodes with degree less than k.N)rr out_degreeadd in_degreedegree)rrnbunchseenr^rks r_low_degree_nodesrns }}uLL0LDz 1KK/LDFJ 0 HHV,LDz -s9B<2B<5B<<3B<3 B<c#"# UR5n[[X!55nU(a[[RR [ URU555nURU5 URU5 UHnU1v M [[X!U55nU(aMUR5(a[R"U5ShvN g[R"U5ShvN gN$N7f)zHelper for filtering components that can't be k-edge-connected. Removes and generates each node with degree less than k. Then generates remaining components where all nodes have degree at least k. N)r#rrnitchain from_iterablemap neighborsdifference_updateremove_nodes_fromrrrr)rrr& singletonsrlr^s r_high_degree_componentsrxs A&q,-J RXX++C Z,HIJ  , J'D&L*189  * }}33A666**1--- 7-s*B4D8.D&D 'DD D DT) returns_graphc## US:a [S5e[[US9nUR5U:a:UR 5H%nUR U/5R 5v M' gU"U5Vs1sH!o@R U5R 5iM# nnU(aUR5nUR5S:XaUv Oy[R"U5n[U5nX:aOURU5 U"U5H1nURUR U5R 55 M3 OUv U(aMggs snf7f)uGeneral algorithm to find all maximal k-edge-connected subgraphs in `G`. Parameters ---------- G : nx.Graph Graph in which all maximal k-edge-connected subgraphs will be found. k : int Yields ------ k_edge_subgraphs : Graph instances that are k-edge-subgraphs Each k-edge-subgraph contains a maximal set of nodes that defines a subgraph of `G` that is k-edge-connected. Notes ----- Implementation of the basic algorithm from [1]_. The basic idea is to find a global minimum cut of the graph. If the cut value is at least k, then the graph is a k-edge-connected subgraph and can be added to the results. Otherwise, the cut is used to split the graph in two and the procedure is applied recursively. If the graph is just a single node, then it is also added to the results. At the end, each result is either guaranteed to be a single node or a subgraph of G that is k-edge-connected. This implementation contains optimizations for reducing the number of calls to max-flow, but there are other optimizations in [1]_ that could be implemented. References ---------- .. [1] Zhou, Liu, et al. (2012) Finding maximal k-edge-connected subgraphs from a large graph. ACM International Conference on Extending Database Technology 2012 480-–491. https://openproceedings.org/2012/conf/edbt/ZhouLYLCL12.pdf Examples -------- >>> from networkx.utils import pairwise >>> paths = [ ... (11, 12, 13, 14, 11, 13, 14, 12), # a 4-clique ... (21, 22, 23, 24, 21, 23, 24, 22), # another 4-clique ... # connect the cliques with high degree but low connectivity ... (50, 13), ... (12, 50, 22), ... (13, 102, 23), ... (14, 101, 24), ... ] >>> G = nx.Graph(it.chain(*[pairwise(path) for path in paths])) >>> sorted(len(k_sg) for k_sg in k_edge_subgraphs(G, k=3)) [1, 1, 1, 4, 4] r r )rN) rrrxrFrr\r#poprminimum_edge_cutr[r$ri) rrfind_ccsr^r]R0G1 cut_edges cut_values rrrs#l 1u233.!4H QGGID**dV$))+ ++31+ 6+B**R.   +B 6 VVX   1 $H++B/III}$$Y/"2,BFF2;;r?//12' " 7sA8E:(E"B2EEr+)rd itertoolsrp functoolsrnetworkxrnetworkx.utilsrr__all__ _dispatchablerrrrr rnrxrr,rrrs A \"O2#O2d\"J1#J1ZZ \"&*#!&*RX2X2v*.2%T&Tr