+ )ic4Rt^RIt^RIt^RIHt^RIHtHt.ROt !RR]P4t ]!R4]!R4]PR444t ]P]P3R l4t]PR 4t]PR 4tR tR tRtR]P3Rlt]!R4]P!RR7R44tR#)z Algorithms for chordal graphs. A graph is chordal if every cycle of length at least 4 has a chord (an edge joining two nodes not adjacent in the cycle). https://en.wikipedia.org/wiki/Chordal_graph N)connected_components)arbitrary_elementnot_implemented_forNetworkXTreewidthBoundExceededc]tRt^tRtRtR#)rzRException raised when a treewidth bound has been provided and it has been exceededN)__name__ __module__ __qualname____firstlineno____doc____static_attributes__rY/var/www/html/photoedit/myenv/lib/python3.14/site-packages/networkx/algorithms/chordal.pyrrsrdirected multigraphch\VP4^8:dR#\\V44^8H#)ulChecks whether G is a chordal graph. A graph is chordal if every cycle of length at least 4 has a chord (an edge joining two nodes not adjacent in the cycle). Parameters ---------- G : graph A NetworkX graph. Returns ------- chordal : bool True if G is a chordal graph and False otherwise. Raises ------ NetworkXNotImplemented The algorithm does not support DiGraph, MultiGraph and MultiDiGraph. Examples -------- >>> e = [ ... (1, 2), ... (1, 3), ... (2, 3), ... (2, 4), ... (3, 4), ... (3, 5), ... (3, 6), ... (4, 5), ... (4, 6), ... (5, 6), ... ] >>> G = nx.Graph(e) >>> nx.is_chordal(G) True Notes ----- The routine tries to go through every node following maximum cardinality search. It returns False when it finds that the separator for any node is not a clique. Based on the algorithms in [1]_. Self loops are ignored. References ---------- .. [1] R. E. Tarjan and M. Yannakakis, Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs, SIAM J. Comput., 13 (1984), pp. 566–579. T)lennodes_find_chordality_breaker)Gs&r is_chordalrs.r 177|q '* +q 00rc.\V4'g\P!R4h\P!V4pVP W4\ 4p\ WAV4pV'dGVwrxp VPV4VFp W8wgK VP W4K \ WAV4pKNV'd[VPV4W,F=p\V\ W,4,4^8XgK+VPV4V# V#)a!Returns the set of induced nodes in the path from s to t. Parameters ---------- G : graph A chordal NetworkX graph s : node Source node to look for induced nodes t : node Destination node to look for induced nodes treewidth_bound: float Maximum treewidth acceptable for the graph H. The search for induced nodes will end as soon as the treewidth_bound is exceeded. Returns ------- induced_nodes : Set of nodes The set of induced nodes in the path from s to t in G Raises ------ NetworkXError The algorithm does not support DiGraph, MultiGraph and MultiDiGraph. If the input graph is an instance of one of these classes, a :exc:`NetworkXError` is raised. The algorithm can only be applied to chordal graphs. If the input graph is found to be non-chordal, a :exc:`NetworkXError` is raised. Examples -------- >>> G = nx.Graph() >>> G = nx.generators.classic.path_graph(10) >>> induced_nodes = nx.find_induced_nodes(G, 1, 9, 2) >>> sorted(induced_nodes) [1, 2, 3, 4, 5, 6, 7, 8, 9] Notes ----- G must be a chordal graph and (s,t) an edge that is not in G. If a treewidth_bound is provided, the search for induced nodes will end as soon as the treewidth_bound is exceeded. The algorithm is inspired by Algorithm 4 in [1]_. A formal definition of induced node can also be found on that reference. Self Loops are ignored References ---------- .. [1] Learning Bounded Treewidth Bayesian Networks. Gal Elidan, Stephen Gould; JMLR, 9(Dec):2699--2731, 2008. http://jmlr.csail.mit.edu/papers/volume9/elidan08a/elidan08a.pdf Input graph is not chordal.) rnx NetworkXErrorGraphadd_edgesetrupdateaddr) rsttreewidth_boundH induced_nodestripletuvwns &&&& rfind_induced_nodesr+\sp a==<==  AJJqEM&q_=G  qW%Av 1 +1A!A=3qt9,-2!!!$   rc#a"V3Rl\S44EFfpVP4^8XdO\P!V4^8d\P!R4h\ VP 44xKg\VP 44p\V4pVPV4V0pV0pV'd\WV4pVPV4VPV4\VPV44V,pVPV4p\V4'd)VPV4We8g\ V4xTpK\P!R4h\ V4xEKi R#5i)aReturns all maximal cliques of a chordal graph. The algorithm breaks the graph in connected components and performs a maximum cardinality search in each component to get the cliques. Parameters ---------- G : graph A NetworkX graph Yields ------ frozenset of nodes Maximal cliques, each of which is a frozenset of nodes in `G`. The order of cliques is arbitrary. Raises ------ NetworkXError The algorithm does not support DiGraph, MultiGraph and MultiDiGraph. The algorithm can only be applied to chordal graphs. If the input graph is found to be non-chordal, a :exc:`NetworkXError` is raised. Examples -------- >>> e = [ ... (1, 2), ... (1, 3), ... (2, 3), ... (2, 4), ... (3, 4), ... (3, 5), ... (3, 6), ... (4, 5), ... (4, 6), ... (5, 6), ... (7, 8), ... ] >>> G = nx.Graph(e) >>> G.add_node(9) >>> cliques = [c for c in chordal_graph_cliques(G)] >>> cliques[0] frozenset({1, 2, 3}) c3b<"TF$pSPV4P4xK& R#5iN)subgraphcopy).0crs& r (chordal_graph_cliques..s' D,Cqajjm  "",Cs,/rN)rnumber_of_nodesrnumber_of_selfloopsr frozensetrrrremove_max_cardinality_noder neighborsr/_is_complete_graph)rC unnumberedr(numberedclique_wanna_benew_clique_wanna_besgsf rchordal_graph_cliquesrBs6\E,@,C D   ! #%%a(1,&&'DEEAGGI& &QWWYJ!!$A   a sH cO)!B!!!$ Q&)!++a.&9H&D#ZZ0%b))'++A..A'88&9O**+HIIO, ,1Es B?FCFc\V4'g\P!R4hRp\P!V4Fp\ V\ V44pK V^, #)a(Returns the treewidth of the chordal graph G. Parameters ---------- G : graph A NetworkX graph Returns ------- treewidth : int The size of the largest clique in the graph minus one. Raises ------ NetworkXError The algorithm does not support DiGraph, MultiGraph and MultiDiGraph. The algorithm can only be applied to chordal graphs. If the input graph is found to be non-chordal, a :exc:`NetworkXError` is raised. Examples -------- >>> e = [ ... (1, 2), ... (1, 3), ... (2, 3), ... (2, 4), ... (3, 4), ... (3, 5), ... (3, 6), ... (4, 5), ... (4, 6), ... (5, 6), ... (7, 8), ... ] >>> G = nx.Graph(e) >>> G.add_node(9) >>> nx.chordal_graph_treewidth(G) 3 References ---------- .. [1] https://en.wikipedia.org/wiki/Tree_decomposition#Treewidth r)rrrrBmaxr)r max_cliquecliques& rchordal_graph_treewidthrHsTZ a==<==J**1-S[1 . >rc\P!V4^8d\P!R4hVP4pV^8dR#VP 4pW^, ,^, pW#8H#)z&Returns True if G is a complete graph.z'Self loop found in _is_complete_graph()T)rr6rr5number_of_edges)rr*e max_edgess& rr;r;+se a 1$HII A1u A!e!I >rc\V4pVFUpV\\W,P44V.,4, pV'gKDW#P43u# R#)z5Given a non-complete graph G, returns a missing edge.N)rlistkeyspop)rrr'missings& r_find_missing_edgerR7sL FE #d1499;/1#566 7{{}% %rcRpVF8p\W,Uu.uF qUV9gK VNK up4pWc8gK4TpTpK: X#uupi)zXReturns a the node in choices that has more connections in G to nodes in wanna_connect. rD)r)rchoices wanna_connect max_numberxynumbermax_cardinality_nodes&&& rr9r9@sSJ <Am);aa<=  J#$   =s A A c \V4^8Xd\P!R4h\V4pVf \ V4pVP V4V0pRpV'd\ WV4pVP V4VPV4\W,4V,pVPV4p\V4'd7\V\V44pWR8d\P!RV 24hK\V4wrWV 3#R#)a/Given a graph G, starts a max cardinality search (starting from s if s is given and from an arbitrary node otherwise) trying to find a non-chordal cycle. If it does find one, it returns (u,v,w) where u,v,w are the three nodes that together with s are involved in the cycle. It ignores any self loops. zGraph has no nodes.ztreewidth_bound exceeded: rDr) rrNetworkXPointlessConceptrrr8r9r r/r;rErrR) rr!r#r=r>current_treewidthr(r?rAr'r)s &&& rrrMs 1v{))*?@@QJy a asH  !! :! Qad)h. ZZ ( b ! ! #$5s?7K L  27701B0CD3(+FQ!9  IrT) returns_graphc a VP4pVUu/uFq"^bK pp\P!V4'dW3#\4pVP 4Uu/uFq"^bK upo \ VP 44p\ \VP 44^R4EFp\VV 3RlR7pVPV4WcV&.pVFp VPW4'dVPV 4K-S V ,p VUu.uFpS V,V 8gKVNK p p\P!VPWV .,4W4'gKVPV 4VPWy34K VFpS V;;,^, uu&K EK VPV4W3#uupiuupiuupi)aReturn a copy of G completed to a chordal graph Adds edges to a copy of G to create a chordal graph. A graph G=(V,E) is called chordal if for each cycle with length bigger than 3, there exist two non-adjacent nodes connected by an edge (called a chord). Parameters ---------- G : NetworkX graph Undirected graph Returns ------- H : NetworkX graph The chordal enhancement of G alpha : Dictionary The elimination ordering of nodes of G Notes ----- There are different approaches to calculate the chordal enhancement of a graph. The algorithm used here is called MCS-M and gives at least minimal (local) triangulation of graph. Note that this triangulation is not necessarily a global minimum. https://en.wikipedia.org/wiki/Chordal_graph References ---------- .. [1] Berry, Anne & Blair, Jean & Heggernes, Pinar & Peyton, Barry. (2004) Maximum Cardinality Search for Computing Minimal Triangulations of Graphs. Algorithmica. 39. 287-298. 10.1007/s00453-004-1084-3. Examples -------- >>> from networkx.algorithms.chordal import complete_to_chordal_graph >>> G = nx.wheel_graph(10) >>> H, alpha = complete_to_chordal_graph(G) c<SV,#r.r)nodeweights&r+complete_to_chordal_graph..s 6$1b )  &? @"a !Azz!##A&"!9%5%5T9PDD%5;;qzz+A*>?FF ''*JJv&"!D 4LA L!'**V 8O9 $-s G GG &G )rr+rBrHrrs)r sysnetworkxrnetworkx.algorithms.componentsrnetworkx.utilsrr__all__NetworkXExceptionr _dispatchablermaxsizer+rBrHr;rRr9rrsrrrr|s ?A R%9%9 Z \"81#!81v03 LL^E-E-P22j &  #' $NZ %E&!Er