Finite maximal drawable extension

Proved

WilliamsonShmoys.planar_maximal_extension__p2m_6a345071_3a17_4f5f_baeb_1c2d7606cf05_a1

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

approximation-algorithmsplanar-graphstreewidth

Every drawable graph on a finite vertex type is contained in a drawable graph maximal under inclusion among drawable supergraphs on that same vertex type.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_maximal_extension__p2m_6a345071_3a17_4f5f_baeb_1c2d7606cf05_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V)
    (hK : WilliamsonShmoys.HasPlanarDrawingOn K) :
    ∃ H : SimpleGraph V,
      K ≤ H ∧
      WilliamsonShmoys.HasPlanarDrawingOn H ∧
      ∀ J : SimpleGraph V, H ≤ J → WilliamsonShmoys.HasPlanarDrawingOn J → J ≤ H := 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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