Maximal predicate-satisfying supergraph on a finite vertex type

Proved

WilliamsonShmoys.finite_simpleGraph_supergraph_maximal__p2m_233ed0bd_e0d7_41b1_8c05_73fad0336759_a1

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

approximation-algorithmsplanar-graphstreewidth

If P holds for a graph H on a finite vertex type, then H has a P-satisfying supergraph that is maximal under graph inclusion.

Preamble

import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.finite_simpleGraph_supergraph_maximal__p2m_233ed0bd_e0d7_41b1_8c05_73fad0336759_a1
    {V : Type} [Fintype V]
    (H : SimpleGraph V) (P : SimpleGraph V → Prop) (hH : P H) :
    exists J : SimpleGraph V,
      And (P J)
        (And (H <= J)
          (forall K : SimpleGraph V,
            P K -> J <= K -> K <= J)) := 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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