Maximal extension of a graph property on a finite vertex type

Proved

WilliamsonShmoys.finite_maximal_simpleGraph__p2m_629021cc_ee60_4d9a_a83c_dbd077e630a9_a3

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

approximation-algorithmsplanar-graphstreewidth

For a finite vertex type, every graph satisfying a property is contained in a graph maximal among those satisfying that property.

Preamble

import Mathlib
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.finite_maximal_simpleGraph__p2m_629021cc_ee60_4d9a_a83c_dbd077e630a9_a3
    {V : Type} [Fintype V]
    (P : SimpleGraph V → Prop)
    (G : SimpleGraph V) (hG : P G) :
    ∃ H : SimpleGraph V, G ≤ H ∧ P H ∧
      ∀ K : SimpleGraph V, H ≤ K → P 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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