Every finite drawable graph has a maximal drawable supergraph

Proved

WilliamsonShmoys.exists_maximal_planar_supergraph__p2m_1634faf2_3284_496c_bfc9_f8ff236eda71_a1

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

approximation-algorithmsplanar-graphstreewidth

For a finite linearly ordered vertex type with at least three vertices, every drawable graph has a drawable supergraph on the same vertices that is maximal among drawable supergraphs on those vertices.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.exists_maximal_planar_supergraph__p2m_1634faf2_3284_496c_bfc9_f8ff236eda71_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V) (hK : HasPlanarDrawingOn K)
    (hV : 3 ≤ Fintype.card V) :
    ∃ L : SimpleGraph V, K ≤ L ∧ HasPlanarDrawingOn L ∧
      ∀ M : SimpleGraph V, L ≤ M → HasPlanarDrawingOn M → M = L := 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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