Existence of a maximal drawable supergraph

Proved

WilliamsonShmoys.exists_maximal_planar_drawing_supergraph__p2m_b6ef5ab1_c465_4724_83b5_1fd8c793e204_a1

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

approximation-algorithmsplanar-graphstreewidth

Every drawable graph on a finite linearly ordered vertex type has a drawable supergraph on the same vertices that is maximal among drawable supergraphs under edge inclusion.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.exists_maximal_planar_drawing_supergraph__p2m_b6ef5ab1_c465_4724_83b5_1fd8c793e204_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V) (hH : HasPlanarDrawingOn H) :
    Exists fun K : SimpleGraph V =>
      And (LE.le H K) (And (HasPlanarDrawingOn K)
        (forall L : SimpleGraph V, LE.le K L -> HasPlanarDrawingOn L -> Eq L K)) := 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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