btj% ddlZddlZddlmZdgZejededdZdS)N)not_implemented_foris_perfect_graphdirected multigraphc tdtjtj|tjtj|D S)uiReturn True if G is a perfect graph, else False. A graph G is perfect if, for every induced subgraph H of G, the chromatic number of H equals the size of the largest clique in H. According to the **Strong Perfect Graph Theorem (SPGT)**: A graph is perfect if and only if neither the graph G nor its complement :math:`\overline{G}` contains an **induced odd hole** — an induced cycle of odd length at least five without chords. Parameters ---------- G : NetworkX Graph The graph to check. Must be a finite, simple, undirected graph. Returns ------- bool True if G is a perfect graph, else False. Notes ----- This function uses a direct approach: cycle enumeration to detect chordless odd cycles in G and :math:`\overline{G}`. This implementation runs in exponential time in the worst case, since the number of chordless cycles can grow exponentially. The perfect-graph recognition problem is theoretically solvable in polynomial time. Chudnovsky *et al.* (2006) proved it can be solved in :math:`O(n^9)` time via a complex structural decomposition [1]_, [2]_. This implementation opts for a direct, transparent check rather than implementing that high-degree polynomial-time decomposition algorithm. See Also -------- is_chordal, is_bipartite : Related checks for specific categories of perfect graphs, such as chordal graphs, and bipartite graphs. chordless_cycles : Used to detect "holes" in the graph References ---------- .. [1] M. Chudnovsky, N. Robertson, P. Seymour, and R. Thomas, *The Strong Perfect Graph Theorem*, Annals of Mathematics, vol. 164, no. 1, pp. 51–229, 2006. https://doi.org/10.4007/annals.2006.164.51 .. [2] M. Chudnovsky, G. Cornuéjols, X. Liu, P. Seymour, and K. Vušković, *Recognizing Berge Graphs*, Combinatorica 25(2): 143–186, 2005. DOI: 10.1007/s00493-005-0003-8 Preprint available at: https://web.math.princeton.edu/~pds/papers/algexp/Bergealg.pdf c3hK|]-}t|dkot|dzdkV.dS)N)len).0cs _/var/www/html/CA-Chatbot/venv/lib/python3.11/site-packages/networkx/algorithms/perfect_graph.py z#is_perfect_graph..DsS  Q1+3q66A:?)any itertoolschainnxchordless_cycles complement)Gs rrr siv   " "B$7 a8H8H$I$I   r)rnetworkxrnetworkx.utils.decoratorsr__all__ _dispatchablerrrrs999999  Z  \""==#"! ===r