Finite downward-closed graph properties have maximal extensions

Proved

WilliamsonShmoys.exists_maximal_extension_of_downward_closed__p2m_138d462c_20bd_43cc_9c59_565c454fd505_a1

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

approximation-algorithmsplanar-graphstreewidth

For a finite vertex type, any graph satisfying a downward-closed property is contained in a supergraph satisfying that property that has no strictly larger supergraph with the property.

Preamble

import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.exists_maximal_extension_of_downward_closed__p2m_138d462c_20bd_43cc_9c59_565c454fd505_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (P : SimpleGraph V → Prop)
    (hdown : ∀ {H K : SimpleGraph V}, H ≤ K → P K → P H)
    (G : SimpleGraph V) (hG : P G) :
    ∃ Q : SimpleGraph V,
      G ≤ Q ∧ P Q ∧
        ∀ R : SimpleGraph V, Q ≤ R → P R → R ≤ Q := 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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