wgc4dZddlZddlZddlmZddlmZmZgdZ GddejZ eded ejd Z ejejfd Zejd Zejd ZdZdZdZdejfdZedejddZy)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_graphceZdZdZy)r zVException raised when a treewidth bound has been provided and it has been exceededN)__name__ __module__ __qualname____doc__`/home/mcse/projects/flask/flask-venv/lib/python3.12/site-packages/networkx/algorithms/chordal.pyr r srr directed multigraphcbt|jdkrytt|dk(S)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 rrrs.r 177|q '* +q 00rct|stjdtj|}|j ||t }t |||}|rF|\}}} |j||D]} | |k7s |j || t |||}|rF|rL|j|||D]3}t|t ||zdk(s!|j||S|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\sp a=<==  AJJq!EM&q!_=G  AqW% !AAv 1a  !+1aA !1 A=3qt9,-2!!!$    rc#KfdtDD]O}|jdk(rItj|dkDrtjdt |j `t|j }t|}|j||h}|h}|rt|||}|j||j|t|j||z}|j|}t|r&|j|||k\s t ||}ntjd|rt |Ryw)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}) c3\K|]#}j|j%ywN)subgraphcopy).0crs r z(chordal_graph_cliques..s" Dqajjm  " Ds),rrN)rnumber_of_nodesrnumber_of_selfloopsr frozensetrr#rremove_max_cardinality_noder% neighborsr3_is_complete_graph)rC unnumberedr-numberedclique_wanna_benew_clique_wanna_besgs` rrrsC\E,@,C D-   ! #%%a(1,&&'DEEAGGI& &QWWYJ!!$A   a sH cO)!ZB!!!$ Q&)!++a.&9H&D#ZZ0%b)'++A../A'88&9O**+HIIO, ,1-s EE,E,ct|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 sWZ a=<==J**1-2S[1 2 >rctj|dkDrtjd|j}|dkry|j }||dz zdz }||k(S)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?+si a 1$HII A1u Aa!e!I >rct|}|D]D}|tt||j|gzz }|s2||jfcSy)z5Given a non-complete graph G, returns a missing edge.N)r#listkeyspop)rrr,missings r_find_missing_edgerT7sS FE &#d1Q499;/1#566 w{{}% %&rcxd}|D]-}t||Dcgc] }||vs| c}}||kDs*|}|}/Scc}w)z`Returns a the node in choices that has more connections in G to nodes in wanna_connect. rG)r)rchoices wanna_connect max_numberxynumbermax_cardinality_nodes rr=r=@sXJ %1>> from networkx.algorithms.chordal import complete_to_chordal_graph >>> G = nx.wheel_graph(10) >>> H, alpha = complete_to_chordal_graph(G) rrGc|Sr2r)nodeweights rz+complete_to_chordal_graph..s 6$<r)keyr8)r4rrr#rrPrangerrHr<has_edgeappendhas_pathr3r%add_edges_from) rr)rcalphachordsunnumbered_nodesiz update_nodesrZy_weight lower_nodesrds @rr r tsT A!" #T1W #E # }}Q%x UF"#'') ,$dAg ,FAGGI 3qwwy>1b )  &? @"a ! 'Azz!Q##A&"!9%5!9PD ;;qzz+A*>?AF ''*JJ1v& '! D 4LA L '*V e8O9 $-s F F> F F)rsysnetworkxrnetworkx.algorithms.componentsrnetworkx.utilsrr__all__NetworkXExceptionr _dispatchablermaxsizerrr r?rTr=rr rrrr|s ?A R%9%9 Z \"81#!81v03 LL^E-E-P22j &  #' $NZ %E&!Er