xhs-note-crawling/python/py/Lib/site-packages/networkx/algorithms/perfect_graph.py

import itertools

import networkx as nx
from networkx.utils.decorators import not_implemented_for

__all__ = ["is_perfect_graph"]


@nx._dispatchable
@not_implemented_for("directed")
@not_implemented_for("multigraph")
def is_perfect_graph(G):
    r"""Return 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
    """

    return not any(
        (len(c) >= 5) and (len(c) % 2 == 1)
        for c in itertools.chain(
            nx.chordless_cycles(G), nx.chordless_cycles(nx.complement(G))
        )
    )