Finite maximal extension for an arbitrary graph property

Proved

WilliamsonShmoys.finite_edge_maximal_extension__p2m_89ca6453_1971_4f8a_b52b_11a113611db9_a1

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

approximation-algorithmsplanar-graphstreewidth

For a finite ordered vertex type and any graph property P, every graph satisfying P has a P supergraph that is maximal among P supergraphs on the same vertex type.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.finite_edge_maximal_extension__p2m_89ca6453_1971_4f8a_b52b_11a113611db9_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (P : SimpleGraph V → Prop) (G : SimpleGraph V) (hG : P G) :
    ∃ L : SimpleGraph V, G.edgeSet ⊆ L.edgeSet ∧ P L ∧
      ∀ K : SimpleGraph V, L.edgeSet ⊆ K.edgeSet → P K →
        K.edgeSet = L.edgeSet := 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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