Finite simple graphs have maximal predicate extensions

Proved

finite_simpleGraph_predicate_maximal_extension__p2m_f94053f9_aa1c_4659_90f3_4794e32e1f9a_a1

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

approximation-algorithmsplanar-graphstreewidth

For any predicate on simple graphs over a finite vertex type, every graph satisfying the predicate has a predicate-satisfying supergraph that is maximal under edge inclusion.

Preamble

import Mathlib

set_option autoImplicit false

Formal statement

theorem finite_simpleGraph_predicate_maximal_extension__p2m_f94053f9_aa1c_4659_90f3_4794e32e1f9a_a1
    {U : Type} [Fintype U]
    (H : SimpleGraph U) (P : SimpleGraph U → Prop) (hP : P H) :
    ∃ K : SimpleGraph U, H ≤ K ∧ P K ∧
      ∀ L : SimpleGraph U, K ≤ L → P L → K = L := 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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