A finite maximal supergraph for any graph predicate

Proved

WilliamsonShmoys.exists_maximal_supergraph_for_predicate__p2m_7202bf9d_a814_4b20_b0a7_1713c12fe256_a1

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

approximation-algorithmsplanar-graphstreewidth

For any predicate on graphs over a finite vertex type, every graph satisfying the predicate has a supergraph satisfying it that is maximal among predicate-satisfying supergraphs.

Preamble

import Mathlib
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.exists_maximal_supergraph_for_predicate__p2m_7202bf9d_a814_4b20_b0a7_1713c12fe256_a1
    {V : Type} [Fintype V] (P : SimpleGraph V → Prop) (G : SimpleGraph V)
    (hG : P G) :
    ∃ H : SimpleGraph V, G ≤ H ∧ P H ∧
      (∀ K : SimpleGraph V, P K → H ≤ 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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