Finite maximal extension for a predicate on simple graphs

Proved

WilliamsonShmoys.finite_maximal_extension__p2m_028c3839_39ea_4f46_a39d_0709e61e420f_a1

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

approximation-algorithmsplanar-graphstreewidth

For any predicate on simple graphs over a finite vertex type, if a graph satisfies the predicate, it has a predicate-satisfying supergraph that is inclusion-maximal among predicate-satisfying graphs.

Preamble

import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.finite_maximal_extension__p2m_028c3839_39ea_4f46_a39d_0709e61e420f_a1
    {V : Type} [Fintype V]
    (P : SimpleGraph V → Prop) (M : SimpleGraph V) (hM : P M) :
    ∃ N : SimpleGraph V, M ≤ N ∧ P N ∧
      ∀ K : SimpleGraph V, 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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