Finite maximal supergraph for an arbitrary property

Proved

WilliamsonShmoys.exists_inclusion_maximal_property_supergraph__p2m_8cd0804a_f792_4057_bb92_319536b24856_a1

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

approximation-algorithmsplanar-graphstreewidth

For any property P on simple graphs over a fixed finite linearly ordered vertex type, every graph satisfying P is contained in a P-satisfying graph that is maximal under graph inclusion among P-satisfying supergraphs.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.exists_inclusion_maximal_property_supergraph__p2m_8cd0804a_f792_4057_bb92_319536b24856_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (P : SimpleGraph V → Prop) (G : SimpleGraph V) (hG : P G) :
    ∃ M : SimpleGraph V,
      G ≤ M ∧ P M ∧
        ∀ N : SimpleGraph V, M ≤ N → P N → N ≤ M := 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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