Finite maximal supergraph for a graph property

Proved

WilliamsonShmoys.exists_maximal_supergraph_for_finite_graph_property__p2m_c7b07ed7_db7c_450f_a64f_c5e3047d491b_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, a graph satisfying the property has an extension satisfying it that is maximal under graph inclusion.

Preamble

import Mathlib

Formal statement

theorem WilliamsonShmoys.exists_maximal_supergraph_for_finite_graph_property__p2m_c7b07ed7_db7c_450f_a64f_c5e3047d491b_a1
    {V : Type} [Fintype V] [DecidableEq V]
    (K : SimpleGraph V) (P : SimpleGraph V → Prop) (hK : P K) :
    ∃ L : SimpleGraph V, P L ∧ K ≤ L ∧
      ∀ M : SimpleGraph V, P M → L ≤ M → M = L := 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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