Finite predicate has an inclusion-maximal supergraph

Proved

WilliamsonShmoys.finitePredicateHasInclusionMaximalSupergraph__p2m_58be6bd2_7b61_44ae_b8f6_153972a57451_a1

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

approximation-algorithmsplanar-graphstreewidth

For any predicate on graphs over a finite ordered vertex type, if the predicate holds for K, then K has a supergraph satisfying the predicate that is maximal under edge inclusion among such supergraphs.

Preamble

import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.finitePredicateHasInclusionMaximalSupergraph__p2m_58be6bd2_7b61_44ae_b8f6_153972a57451_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V) [DecidableRel K.Adj]
    (P : SimpleGraph V → Prop) (hP : P K) :
    ∃ (H : SimpleGraph V), K ≤ H ∧ P H ∧
      ∀ (J : SimpleGraph V), H ≤ J → P J → J ≤ 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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