Finite maximal extension for a graph property

Proved

WilliamsonShmoys.exists_maximal_extension_finite__p2m_89fd7ec5_d704_44ae_9887_a6594412388a_a3

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

approximation-algorithmsplanar-graphstreewidth

For any property of simple graphs on a finite linearly ordered vertex type, a graph satisfying the property has a supergraph satisfying it that is maximal among such supergraphs.

Preamble

import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.exists_maximal_extension_finite__p2m_89fd7ec5_d704_44ae_9887_a6594412388a_a3
    {V : Type} [Fintype V] [LinearOrder V]
    (P : SimpleGraph V -> Prop)
    (K : SimpleGraph V)
    (hK : P K) :
    Exists fun H : SimpleGraph V => And (LE.le K H)
      (And (P H) (∀ J : SimpleGraph V, LE.le H J -> P J -> LE.le J 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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