Finite graph properties admit maximal extensions

Proved

WilliamsonShmoys.finite_graph_property_maximal_extension__p2m_f1c70f2e_734a_4ab3_9ef1_4b1e0b3c2cfc_a1

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

approximation-algorithmsplanar-graphstreewidth

For any property of graphs on a finite linearly ordered vertex type, every graph satisfying the property has a supergraph satisfying it that is maximal under graph inclusion.

Preamble

import Mathlib

open SimpleGraph

Formal statement

theorem WilliamsonShmoys.finite_graph_property_maximal_extension__p2m_f1c70f2e_734a_4ab3_9ef1_4b1e0b3c2cfc_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (P : SimpleGraph V → Prop) (H : SimpleGraph V) (hH : P H) :
    ∃ K : SimpleGraph V, H ≤ K ∧ P K ∧
      ∀ L : SimpleGraph V, K ≤ L → P L → L = K := 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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