\Mh[4dZddlZddlZddlmZddlmZmZgdZ Gddej Z eded ej d Z ej ejfd Zej d Zej d ZdZdZdZdejfdZedej ddZdS)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_graphceZdZdZdS)r zVException raised when a treewidth bound has been provided and it has been exceededN)__name__ __module__ __qualname____doc__[/var/www/html/test/jupyter/venv/lib/python3.11/site-packages/networkx/algorithms/chordal.pyr r srr directed multigraphcvt|jdkrdStt|dkS)uChecks 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 rrrs9r 17||qt '** + +q 00rcDt|stjdtj|}|||t }t |||}|rO|\}}} |||D]} | |kr||| t |||}|O|r`||||D]B}t|t ||zdkr||nC|S)aReturns 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\sJp a==><===  AJJq!EEM&q!_==G B AqW%%% ! !AAvv 1a   *1aAA B!1  A=3qt99,--22!!!$$$3 rc#jKfdtDD]}|dkrPtj|dkrtjdt |Vkt|}t|}| ||h}|h}|rt|||}| || |t| ||z}| |}t|r/| |||kst |V|}ntjd|t |VdS)aUReturns all maximal cliques of a chordal graph. The algorithm breaks the graph in connected components and performs a maximum cardinality search in each component to get the cliques. Parameters ---------- G : graph A NetworkX graph Yields ------ frozenset of nodes Maximal cliques, each of which is a frozenset of nodes in `G`. The order of cliques is arbitrary. Raises ------ NetworkXError The algorithm does not support DiGraph, MultiGraph and MultiDiGraph. The algorithm can only be applied to chordal graphs. If the input graph is found to be non-chordal, a :exc:`NetworkXError` is raised. Examples -------- >>> e = [ ... (1, 2), ... (1, 3), ... (2, 3), ... (2, 4), ... (3, 4), ... (3, 5), ... (3, 6), ... (4, 5), ... (4, 6), ... (5, 6), ... (7, 8), ... ] >>> G = nx.Graph(e) >>> G.add_node(9) >>> cliques = [c for c in chordal_graph_cliques(G)] >>> cliques[0] frozenset({1, 2, 3}) c3fK|]+}|V,dSN)subgraphcopy).0crs r z(chordal_graph_cliques..s9 D Dqajjmm  "" D D D D D DrrrN)rnumber_of_nodesrnumber_of_selfloopsr frozensetrr#rremove_max_cardinality_noder% neighborsr3_is_complete_graph)rC unnumberedr-numberedclique_wanna_benew_clique_wanna_besgs` rrrs\E D D D,@,C,C D D D--    ! # #%a((1,,&'DEEEAGGII&& & & & &QWWYYJ!!$$A   a sH cO J)!ZBB!!!$$$ Q&)!++a..&9&9H&D#ZZ00%b))J'++A..../AA'88888&9OO*+HIII JO,, , , , ,1--rct|stjdd}tj|D]}t |t |} |dz S)aReturns the treewidth of the chordal graph G. Parameters ---------- G : graph A NetworkX graph Returns ------- treewidth : int The size of the largest clique in the graph minus one. Raises ------ NetworkXError The algorithm does not support DiGraph, MultiGraph and MultiDiGraph. The algorithm can only be applied to chordal graphs. If the input graph is found to be non-chordal, a :exc:`NetworkXError` is raised. Examples -------- >>> e = [ ... (1, 2), ... (1, 3), ... (2, 3), ... (2, 4), ... (3, 4), ... (3, 5), ... (3, 6), ... (4, 5), ... (4, 6), ... (5, 6), ... (7, 8), ... ] >>> G = nx.Graph(e) >>> G.add_node(9) >>> nx.chordal_graph_treewidth(G) 3 References ---------- .. [1] https://en.wikipedia.org/wiki/Tree_decomposition#Treewidth rr8)rrr rmaxr)r max_cliquecliques rr r seZ a==><===J*1--22S[[11 >rctj|dkrtjd|}|dkrdS|}||dz zdz }||kS)z&Returns True if G is a complete graph.rz'Self loop found in _is_complete_graph()rTr8)rr:r r9number_of_edges)rr/e max_edgess rr?r?+sv a  1$$HIII A1uut Aa!e!I >rct|}|D]W}|tt|||gzz }|r||fcSXdS)z5Given a non-complete graph G, returns a missing edge.N)r#listkeyspop)rrr,missings r_find_missing_edgerT7sy FFE &&#d1Q499;;//1#5666  &w{{}}% % % % &&&rcld}|D]-}tfd||D}||kr|}|}.|S)z`Returns a the node in choices that has more connections in G to nodes in wanna_connect. rGcg|]}|v| Srr)r5y wanna_connects r z)_max_cardinality_node..Fs#<<>> from networkx.algorithms.chordal import complete_to_chordal_graph >>> G = nx.wheel_graph(10) >>> H, alpha = complete_to_chordal_graph(G) rrGc|Sr2r)nodeweights rz+complete_to_chordal_graph..s 6$<r)keyc,g|]}|k|Srr)r5rerfy_weights rrYz-complete_to_chordal_graph..s.!9P9PD9P9P9Prr8)r4dictfromkeysrrr#rrPrangerrHr<has_edgeappendhas_pathr3r%add_edges_from) rr)alphachordsunnumbered_nodesiz update_nodesrW lower_nodesrerfrjs @@rr r tsT A MM!Q  E }Q%x UUF ]]17799a ( (FAGGII 3qwwyy>>1b ) )  &?&?&?&? @ @ @"""a ! ' 'Azz!Q '##A&&&&"!9%5 ;qzz+A*>??AFF' ''***JJ1v&&&   D 4LLLA LLLL V e8Or)rsysnetworkxrnetworkx.algorithms.componentsrnetworkx.utilsrr__all__NetworkXExceptionr _dispatchablermaxsizerrr r?rTr=rr rrrrs ??????AAAAAAAA   R%9 Z  \""8181#"! 81v03 LLLL^E-E-E-P222j   &&&    #' $$$$NZ  %%%EE&%! EEEr