Finite choice of a maximal graph satisfying a property

Proved

WilliamsonShmoys.exists_maximal_supergraph_by_finite_choice__p2m_8945f9a8_f1f1_4069_b3b7_48c3e7c6f526_a1

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

approximation-algorithmsplanar-graphstreewidth

For a finite ordered vertex type, any graph P satisfying a property Q has a Q-satisfying supergraph that is inclusion maximal among Q-satisfying supergraphs of P.

Preamble

import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.exists_maximal_supergraph_by_finite_choice__p2m_8945f9a8_f1f1_4069_b3b7_48c3e7c6f526_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (P : SimpleGraph V)
    (Q : SimpleGraph V → Prop)
    (hP : Q P) :
    ∃ H : SimpleGraph V,
      P ≤ H ∧
      Q H ∧
      (∀ K : SimpleGraph V, Q K → H ≤ K → K ≤ H) := 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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