Maximal extension in a finite family of graphs

Proved

finite_exists_maximal_extension__p2m_2521a354_9295_4d68_a93a_2d2a5f622a67_a1

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

approximation-algorithmsplanar-graphstreewidth

For a finite vertex type, any graph satisfying a property has an extension satisfying that property which is maximal among its extensions under graph inclusion.

Preamble

import Mathlib
set_option autoImplicit false

Formal statement

theorem finite_exists_maximal_extension__p2m_2521a354_9295_4d68_a93a_2d2a5f622a67_a1
    {V : Type} [Fintype V] [DecidableEq V]
    (P : SimpleGraph V → Prop)
    (M : SimpleGraph V) (hM : P M) :
    Exists fun N : SimpleGraph V => And (LE.le M N) (And (P N)
      (forall K : SimpleGraph V, LE.le N K -> P K -> K = N)) := 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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