+ j*Rt^RIHt^RIHt^RIHt^RIt^RI H t ^RI H t ] t R.t] !R4]PRR l44tR tR tR tR tR#)z, Moody and White algorithm for k-components ) defaultdict) combinations) itemgetterN) edmonds_karp)not_implemented_for k_componentsdirectedc \\4pVf\p\P!V4F8p\ V4p\ V4^8gK V^,PV4K: \P!V4Uu.uFqPPV4NK ppVF8p\ V4p\ V4^8gK V^,PV4K: VEFFp \ V 4^8:dK\P!WR7p V ^8d!W*,P\ V 44\\P!WVR74p V \WV 43.p V 'gKV R,wr\V4pV PV4p\P!VVR7pVV 8d)V^8d"VV,P\ V44\\P!VVVR74p V 'd!V PV\VV V434KK \!V4#uupi \dT P4Ki;i)a Returns the k-component structure of a graph G. A `k`-component is a maximal subgraph of a graph G that has, at least, node connectivity `k`: we need to remove at least `k` nodes to break it into more components. `k`-components have an inherent hierarchical structure because they are nested in terms of connectivity: a connected graph can contain several 2-components, each of which can contain one or more 3-components, and so forth. Parameters ---------- G : NetworkX graph flow_func : function Function to perform the underlying flow computations. Default value :meth:`edmonds_karp`. This function performs better in sparse graphs with right tailed degree distributions. :meth:`shortest_augmenting_path` will perform better in denser graphs. Returns ------- k_components : dict Dictionary with all connectivity levels `k` in the input Graph as keys and a list of sets of nodes that form a k-component of level `k` as values. Raises ------ NetworkXNotImplemented If the input graph is directed. Examples -------- >>> # Petersen graph has 10 nodes and it is triconnected, thus all >>> # nodes are in a single component on all three connectivity levels >>> G = nx.petersen_graph() >>> k_components = nx.k_components(G) >>> sorted(k_components) [1, 2, 3] >>> all(comps == [set(G)] for comps in k_components.values()) True Notes ----- Moody and White [1]_ (appendix A) provide an algorithm for identifying k-components in a graph, which is based on Kanevsky's algorithm [2]_ for finding all minimum-size node cut-sets of a graph (implemented in :meth:`all_node_cuts` function): 1. Compute node connectivity, k, of the input graph G. 2. Identify all k-cutsets at the current level of connectivity using Kanevsky's algorithm. 3. Generate new graph components based on the removal of these cutsets. Nodes in a cutset belong to both sides of the induced cut. 4. If the graph is neither complete nor trivial, return to 1; else end. This implementation also uses some heuristics (see [3]_ for details) to speed up the computation. See also -------- node_connectivity all_node_cuts biconnected_components : special case of this function when k=2 k_edge_components : similar to this function, but uses edge-connectivity instead of node-connectivity References ---------- .. [1] Moody, J. and D. White (2003). Social cohesion and embeddedness: A hierarchical conception of social groups. American Sociological Review 68(1), 103--28. http://www2.asanet.org/journals/ASRFeb03MoodyWhite.pdf .. [2] Kanevsky, A. (1993). Finding all minimum-size separating vertex sets in a graph. Networks 23(6), 533--541. http://onlinelibrary.wiley.com/doi/10.1002/net.3230230604/abstract .. [3] Torrents, J. and F. Ferraro (2015). Structural Cohesion: Visualization and Heuristics for Fast Computation. https://arxiv.org/pdf/1503.04476v1 ) flow_func)kr )rlistdefault_flow_funcnxconnected_componentssetlenappendbiconnected_componentssubgraphnode_connectivity all_node_cuts_generate_partitionnext StopIterationpop_reconstruct_k_components)Gr r componentcompc bicomponents bicomponentbicompBr cutsstackparent_k partitionnodesCthis_ks&& m/home/luca/screen-checking/.venv/lib/python3.14/site-packages/networkx/algorithms/connectivity/kcomponents.pyrrs|t$L% ,,Q/ 9~ t9q= O " "4 ( 0 ,.+D+DQ+GH+GaJJqM+GLH# [! v;? O " "6 * $  q6Q;    8 q5 O " "3q6 *B$$QyAB(!456e$)"I !X YJJu%--a9EH$! (//A7B,,Q&INOLL&*=av*N!OP%< %\ 22II4!   sH59B+H::IIc#paa"\P!4p\\V44oVP S4VP VV3Rl\ S^444\P!V4F0p\P!VUu.uF pSV,NK up!xK2 R#uupi5i)aWMerge sets that share k or more elements. See: http://rosettacode.org/wiki/Set_consolidation The iterative python implementation posted there is faster than this because of the overhead of building a Graph and calling nx.connected_components, but it's not clear for us if we can use it in NetworkX because there is no licence for the code. c3|<"TF1wr\SV,SV,,4S8gK,W3xK3 R#5iNr).0uvr r)s& r, _consolidate..s5141SqE!H9L5MQR5R1s)< <N) rGraphdict enumerateadd_nodes_fromadd_edges_fromrrrunion)setsr rrnr)s&f @r, _consolidater>s  A 4 !EU'q1,,Q/ iiI6Iq%((I67706sBB6B1 $B6c #a a "VP4o VUu.uFp\V4NK pp.p\4p\V4.pV'Ed=VP4pVFpW78gK \\P !VP Ws, 444p\V4^8gKQVFo \S VU u0uFSp \;QJd%V 3RlW ,4F 'gK RM RM!V 3RlW ,44'gKQV kKU up ,4p W9gK{VPV 4VP\V 44K EK \V4S 8gEK1VPV4EKE\\WB^,44p \;QJdV 3RlV 4F 'gK RM RM!V 3RlV 44'd TRjxL R#T RjxL R#uupiuup iLL5i)aGenerate parts of G induced by its minimum-size node cutsets. Following Moody and White (appendix A, step 3), cutsets are applied one at a time: removing a cutset splits the current set of nodes into two or more connected components, and each component plus the cutset that induced it becomes a new candidate part. Every j-component of ``G`` with ``j > k`` is preserved intact in at least one generated part, because removing ``k < j`` nodes cannot disconnect it. Applying cutsets one at a time is necessary for correctness: removing the union of all cutsets in a single pass can drop cutset nodes whose neighbors all belong to other cutsets, silently losing k-components. c3,<"TF qS9xK R#5ir/)r1r3rs& r,r4&_generate_partition..s0NAisTFc3@<"TFp\V4S8HxK R#5ir/r0)r1partorders& r,r4rBs 7,$3t9 ,sN) rErrr rrrr frozensetanyaddrr>)rr%r cutpartsseenr&r) componentsr=child consolidatedrrEs&&& @@r,rrs GGIE $ %CHD % E 5D VHE % C;b55ajj6MNOJ:"!+I &!&)OcSS0N0NSSS0N0N-N11cOPE( SZ0",%*5zE! 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