Finite simple graph families have an inclusion-maximal member

Proved

WilliamsonShmoys.finite_simpleGraph_has_maximal_element__p2m_21abd8cf_71a5_4301_9a22_948ffb9f2078_a1

by Eyal1990 · Nov 1, 2026 · Mathlib 0df444a (Lean v4.33.1)

approximation-algorithmsplanar-graphstreewidth

For a finite linearly ordered vertex type, every nonempty family of simple graphs on that type has a member maximal under graph inclusion. This packages the finite maximization step; the target applies it to drawable supergraphs of H. The boundary case of an empty family is excluded by hP, while the single-member case is immediate.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.finite_simpleGraph_has_maximal_element__p2m_21abd8cf_71a5_4301_9a22_948ffb9f2078_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (P : SimpleGraph V → Prop)
    (hP : ∃ K : SimpleGraph V, P K) :
    ∃ K : SimpleGraph V, P K ∧
      ∀ L : SimpleGraph V, P L → K ≤ L → L ≤ K := by sorry

Source

David P. Williamson and David B. Shmoys, The Design of Approximation Algorithms, Cambridge University Press, 2011, author electronic manuscript, Section 10.2, proof of Theorem 10.11, pp. 270-272. https://doi.org/10.1017/CBO9780511921735. Bounded tree-width of the pieces: Robertson–Seymour (planar graphs of radius r have tree-width at most 3r+1) / Bodlaender (k-outerplanar graphs have tree-width at most 3k-1).

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