Finite maximal extension for a graph property

Proved

WilliamsonShmoys.finite_property_maximal_extension__p2m_d19fe924_8be5_4905_9de7_834ee0c8d168_a1

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

approximation-algorithmsplanar-graphstreewidth

For any property of simple graphs on a finite vertex type, every graph satisfying the property is contained in an edge-maximal graph satisfying it.

Preamble

import Mathlib

Formal statement

theorem WilliamsonShmoys.finite_property_maximal_extension__p2m_d19fe924_8be5_4905_9de7_834ee0c8d168_a1
    {V : Type} [Fintype V]
    (P : SimpleGraph V → Prop)
    (H : SimpleGraph V) [DecidableRel H.Adj]
    (hH : P H) :
    ∃ (K : SimpleGraph V) (dK : DecidableRel K.Adj),
      (letI : DecidableRel K.Adj := dK
       P K ∧
       (∀ (L : SimpleGraph V) [DecidableRel L.Adj],
         P L →
         K.edgeFinset ⊆ L.edgeFinset →
         K.edgeFinset = L.edgeFinset) ∧
       H.edgeFinset ⊆ K.edgeFinset) := 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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