hc4SrSSKrSSKrSSKJr SSKJrJr /SQr "SS\R5r \"S5\"S 5\RS 555r \R\R4S j5r\RS 5r\RS 5rSrSrSrS\R4Sjr\"S5\R"SS9S55rg)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_for) is_chordalfind_induced_nodeschordal_graph_cliqueschordal_graph_treewidthNetworkXTreewidthBoundExceededcomplete_to_chordal_graphc\rSrSrSrSrg)r zRException raised when a treewidth bound has been provided and it has been exceededN)__name__ __module__ __qualname____firstlineno____doc____static_attributes__rM/var/www/html/env/lib/python3.13/site-packages/networkx/algorithms/chordal.pyr r srr directed multigraphcd[UR5S::ag[[U55S:H$)ulChecks whether G is a chordal graph. A graph is chordal if every cycle of length at least 4 has a chord (an edge joining two nodes not adjacent in the cycle). Parameters ---------- G : graph A NetworkX graph. Returns ------- chordal : bool True if G is a chordal graph and False otherwise. Raises ------ NetworkXNotImplemented The algorithm does not support DiGraph, MultiGraph and MultiDiGraph. Examples -------- >>> e = [ ... (1, 2), ... (1, 3), ... (2, 3), ... (2, 4), ... (3, 4), ... (3, 5), ... (3, 6), ... (4, 5), ... (4, 6), ... (5, 6), ... ] >>> G = nx.Graph(e) >>> nx.is_chordal(G) True Notes ----- The routine tries to go through every node following maximum cardinality search. It returns False when it finds that the separator for any node is not a clique. Based on the algorithms in [1]_. Self loops are ignored. References ---------- .. [1] R. E. Tarjan and M. Yannakakis, Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs, SIAM J. Comput., 13 (1984), pp. 566–579. Tr)lennodes_find_chordality_breaker)Gs rrrs.r 177|q '* +q 00rc[U5(d[R"S5e[R"U5nUR X5 [ 5n[ XAU5nU(aLUupxn URU5 UHn X:wdM UR X5 M [ XAU5nU(aMLU(aMURU5 XH4n[U[ X5-5S:XdM"URU5 U$ U$)a!Returns the set of induced nodes in the path from s to t. Parameters ---------- G : graph A chordal NetworkX graph s : node Source node to look for induced nodes t : node Destination node to look for induced nodes treewidth_bound: float Maximum treewidth acceptable for the graph H. The search for induced nodes will end as soon as the treewidth_bound is exceeded. Returns ------- induced_nodes : Set of nodes The set of induced nodes in the path from s to t in G Raises ------ NetworkXError The algorithm does not support DiGraph, MultiGraph and MultiDiGraph. If the input graph is an instance of one of these classes, a :exc:`NetworkXError` is raised. The algorithm can only be applied to chordal graphs. If the input graph is found to be non-chordal, a :exc:`NetworkXError` is raised. Examples -------- >>> G = nx.Graph() >>> G = nx.generators.classic.path_graph(10) >>> induced_nodes = nx.find_induced_nodes(G, 1, 9, 2) >>> sorted(induced_nodes) [1, 2, 3, 4, 5, 6, 7, 8, 9] Notes ----- G must be a chordal graph and (s,t) an edge that is not in G. If a treewidth_bound is provided, the search for induced nodes will end as soon as the treewidth_bound is exceeded. The algorithm is inspired by Algorithm 4 in [1]_. A formal definition of induced node can also be found on that reference. Self Loops are ignored References ---------- .. [1] Learning Bounded Treewidth Bayesian Networks. Gal Elidan, Stephen Gould; JMLR, 9(Dec):2699--2731, 2008. http://jmlr.csail.mit.edu/papers/volume9/elidan08a/elidan08a.pdf Input graph is not chordal.) rnx NetworkXErrorGraphadd_edgesetrupdateaddr) rsttreewidth_boundH induced_nodestripletuvwns rrr\sp a==<==  AJJqEM&q_=G  qW%Av 1 +1A '!A=3qt9,-2!!!$   rc# ^# U4Sj[T55GHenUR5S:XaM[R"U5S:a[R"S5e[ UR 55v Me[UR 55n[U5nURU5 U1nU1nU(a[XU5nURU5 URU5 [URU55U-nURU5n[U5(a&URU5 Xe:d [ U5v UnO[R"S5eU(aM[ U5v GMh g7f)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># UH$nTRU5R5v M& g7fN)subgraphcopy).0crs r (chordal_graph_cliques..s' D,Cqajjm  "",Cs,/rr N)rnumber_of_nodesr"number_of_selfloopsr# frozensetrr&rremove_max_cardinality_noder( neighborsr6_is_complete_graph)rC unnumberedr0numberedclique_wanna_benew_clique_wanna_besgs` rrrs?\E,@,C D   ! #%%a(1,&&'DEEAGGI& &QWWYJ!!$A   a sH cO)!B!!!$ Q&)!++a.&9H&D#ZZ0%b))'++A..A'88&9O**+HII*O, ,1Es E,F1Fc[U5(d[R"S5eSn[R"U5Hn[ U[ U55nM US- $)a(Returns the treewidth of the chordal graph G. Parameters ---------- G : graph A NetworkX graph Returns ------- treewidth : int The size of the largest clique in the graph minus one. Raises ------ NetworkXError The algorithm does not support DiGraph, MultiGraph and MultiDiGraph. The algorithm can only be applied to chordal graphs. If the input graph is found to be non-chordal, a :exc:`NetworkXError` is raised. Examples -------- >>> e = [ ... (1, 2), ... (1, 3), ... (2, 3), ... (2, 4), ... (3, 4), ... (3, 5), ... (3, 6), ... (4, 5), ... (4, 6), ... (5, 6), ... (7, 8), ... ] >>> G = nx.Graph(e) >>> G.add_node(9) >>> nx.chordal_graph_treewidth(G) 3 References ---------- .. [1] https://en.wikipedia.org/wiki/Tree_decomposition#Treewidth r r<)rr"r#rmaxr)r max_cliquecliques rr r sTZ a==<==J**1-S[1 . >rc[R"U5S:a[R"S5eUR5nUS:agUR 5nXS- -S- nX#:H$)z&Returns True if G is a complete graph.rz'Self loop found in _is_complete_graph()r!Tr<)r"r>r#r=number_of_edges)rr2e max_edgess rrCrC+se a 1$HII A1u A!e!I >rc[U5nUHHnU[[XR55U/-5- nU(dM7X#R54s $ g)z5Given a non-complete graph G, returns a missing edge.N)r&listkeyspop)rrr/missings r_find_missing_edgerX7sL FE #d1499;/1#566 7{{}% %rcSnUH3n[XVs/sH oUU;dM UPM sn5nXc:dM/UnUnM5 W$s snf)zXReturns a the node in choices that has more connections in G to nodes in wanna_connect. rK)r)rchoices wanna_connect max_numberxynumbermax_cardinality_nodes rrArA@sQJ <Am);a<=  J#$   =s > > c[U5S:Xa[R"S5e[U5nUc [ U5nUR U5 U1nSnU(a[ XU5nUR U5 URU5 [X5U-nURU5n[U5(a4[U[U55nXR:a[R"SU35eO[U5upXU 4$U(aMg)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. rzGraph has no nodes.rKztreewidth_bound exceeded: r) rr"NetworkXPointlessConceptr&rr@rAr(r6rCrLr rX) rr)r+rErFcurrent_treewidthr0rGrIr/r1s rrrMs 1v{))*?@@QJy a asH  !! :! Qad)h. ZZ ( b ! ! #$5s?7K L  27701B0CD3(+FQ!9 # *$ rT) returns_graphc n^ UR5nUVs0sHo"S_M nn[R"U5(aX4$[5nUR 5Vs0sHo"S_M snm [ UR 55n[ [UR 55SS5Hn[UU 4SjS9nURU5 XcU'/nUHn URX5(aURU 5 M,T U n UVs/sHnT UU :dMUPM n n[R"URXU /-5X5(dMURU 5 URXy45 M UHnT U==S- ss'M M URU5 X4$s snfs snfs snf)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) rrKc>TU$r5r)nodeweights r+complete_to_chordal_graph..s 6$1b )  &? @"a !Azz!##A&"!9%5%5T9PD%5;;qzz+A*>?FF ''*JJv&"!D 4LA L!'**V 8O9 $-s F( F- F2F2)rsysnetworkxr"networkx.algorithms.componentsrnetworkx.utilsrr__all__NetworkXExceptionr _dispatchablermaxsizerrr rCrXrArr rrrrs ?A R%9%9 Z \"81#!81v03 LL^E-E-P22j &  #' $NZ %E&!Er