Finite maximal drawable extension

Proved

WilliamsonShmoys.planar_drawing_finite_maximal_extension__p2m_c5c73f75_8c1e_44ef_9deb_26150704a428_a1

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

approximation-algorithmsplanar-graphstreewidth

Every drawable graph on a finite ordered vertex type is contained in a drawable supergraph maximal among drawable supergraphs on the same vertex type.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_drawing_finite_maximal_extension__p2m_c5c73f75_8c1e_44ef_9deb_26150704a428_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V)
    (hK : WilliamsonShmoys.HasPlanarDrawingOn K) :
    Exists fun H : SimpleGraph V =>
      And (K <= H) (And (WilliamsonShmoys.HasPlanarDrawingOn H)
        (forall J : SimpleGraph V, H <= J ->
          WilliamsonShmoys.HasPlanarDrawingOn J -> H = J)) := 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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