Finite maximal extension for a graph property

Proved

WilliamsonShmoys_finite_maximal_drawable_extension__p2m_646dfb24_703c_493f_8075_444718d53d7b_a1

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

approximation-algorithmsplanar-graphstreewidth

On a finite ordered vertex type, any graph satisfying a property has a property-satisfying supergraph that is inclusion-maximal among such supergraphs; its edge count is at least that of the starting graph.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys_finite_maximal_drawable_extension__p2m_646dfb24_703c_493f_8075_444718d53d7b_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V) (P : SimpleGraph V → Prop) (hK : P K) :
    ∃ H : SimpleGraph V,
      K ≤ H ∧ P H ∧
      (∀ G : SimpleGraph V, H ≤ G → P G → G ≤ H) ∧
      (letI : DecidableRel K.Adj := Classical.decRel K.Adj
       letI : DecidableRel H.Adj := Classical.decRel H.Adj
       K.edgeFinset.card ≤ H.edgeFinset.card) := 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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