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import itertools
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import networkx as nx
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from networkx.utils.decorators import not_implemented_for
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__all__ = ["is_perfect_graph"]
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@nx._dispatchable
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@not_implemented_for("directed")
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@not_implemented_for("multigraph")
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def is_perfect_graph(G):
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r"""Return True if G is a perfect graph, else False.
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A graph G is perfect if, for every induced subgraph H of G, the chromatic
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number of H equals the size of the largest clique in H.
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According to the **Strong Perfect Graph Theorem (SPGT)**:
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A graph is perfect if and only if neither the graph G nor its complement
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:math:`\overline{G}` contains an **induced odd hole** — an induced cycle of
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odd length at least five without chords.
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Parameters
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----------
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G : NetworkX Graph
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The graph to check. Must be a finite, simple, undirected graph.
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Returns
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-------
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bool
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True if G is a perfect graph, else False.
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Notes
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-----
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This function uses a direct approach: cycle enumeration to detect
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chordless odd cycles in G and :math:`\overline{G}`. This implementation
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runs in exponential time in the worst case, since the number of chordless
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cycles can grow exponentially.
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The perfect-graph recognition problem is theoretically solvable in
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polynomial time. Chudnovsky *et al.* (2006) proved it can be solved in
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:math:`O(n^9)` time via a complex structural decomposition [1]_, [2]_.
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This implementation opts for a direct, transparent check rather than
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implementing that high-degree polynomial-time decomposition algorithm.
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See Also
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--------
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is_chordal, is_bipartite :
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Related checks for specific categories of perfect graphs, such as chordal
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graphs, and bipartite graphs.
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chordless_cycles :
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Used to detect "holes" in the graph
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References
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----------
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.. [1] M. Chudnovsky, N. Robertson, P. Seymour, and R. Thomas,
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*The Strong Perfect Graph Theorem*,
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Annals of Mathematics, vol. 164, no. 1, pp. 51–229, 2006.
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https://doi.org/10.4007/annals.2006.164.51
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.. [2] M. Chudnovsky, G. Cornuéjols, X. Liu, P. Seymour, and K. Vušković,
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*Recognizing Berge Graphs*,
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Combinatorica 25(2): 143–186, 2005.
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DOI: 10.1007/s00493-005-0003-8
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Preprint available at:
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https://web.math.princeton.edu/~pds/papers/algexp/Bergealg.pdf
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"""
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return not any(
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(len(c) >= 5) and (len(c) % 2 == 1)
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for c in itertools.chain(
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nx.chordless_cycles(G), nx.chordless_cycles(nx.complement(G))
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)
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)
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