Existence of an inclusion-maximal drawable extension

Proved

WilliamsonShmoys.drawable_extension_maximal__p2m_3c50d639_b449_4a2f_a283_3dfd9e523102_a1

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

approximation-algorithmsplanar-graphstreewidth

Every drawable graph on a finite ordered vertex type has an inclusion-maximal drawable supergraph on the same vertex type. This removes the finite optimization step from the parent. Prove it by considering the finite set of simple graphs on V that contain P and are drawable, then choosing one with maximum edge count; any drawable strict extension would contradict maximality. The set is nonempty because it contains P. No vertex-count lower bound is needed for this step.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib
open WilliamsonShmoys

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.drawable_extension_maximal__p2m_3c50d639_b449_4a2f_a283_3dfd9e523102_a1 {V : Type} [Fintype V] [LinearOrder V] (P : SimpleGraph V) [DecidableRel P.Adj] (hP : HasPlanarDrawingOn P) : ∃ Q : SimpleGraph V, ∃ hQdec : DecidableRel Q.Adj, P ≤ Q ∧ HasPlanarDrawingOn Q ∧ ∀ (R : SimpleGraph V) (hRdec : DecidableRel R.Adj), Q ≤ R → HasPlanarDrawingOn R → R ≤ Q := 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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