Finite maximal extension for a downward-closed graph property

Proved

exists_maximal_supergraph_of_downward_closed__p2m_5641e734_866b_4e6a_9d69_40c2e1f45d44_a1

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

approximation-algorithmsplanar-graphstreewidth

For a finite vertex type, every graph satisfying a downward-closed property is contained in a satisfying supergraph that is maximal among satisfying supergraphs.

Preamble

import Mathlib

Formal statement

theorem exists_maximal_supergraph_of_downward_closed__p2m_5641e734_866b_4e6a_9d69_40c2e1f45d44_a1
    {V : Type} [Fintype V] [DecidableEq V]
    (P : SimpleGraph V → Prop)
    (hdown : ∀ {A B : SimpleGraph V}, A ≤ B → P B → P A)
    (G : SimpleGraph V) (hG : P G) :
    ∃ K : SimpleGraph V, G ≤ K ∧ P K ∧
      ∀ H : SimpleGraph V, K ≤ H → P H → H ≤ 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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