Every drawable graph has a maximal drawable supergraph

Proved

WilliamsonShmoys.drawable_planar_supergraph_maximal__p2m_5303ee26_08e8_4d11_9742_cd82622b8993_a1

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

approximation-algorithmsplanar-graphstreewidth

Any drawable graph on a finite vertex type with at least three vertices has an inclusion-maximal drawable supergraph on that same vertex type.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.drawable_planar_supergraph_maximal__p2m_5303ee26_08e8_4d11_9742_cd82622b8993_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (M : SimpleGraph V)
    (hM : HasPlanarDrawingOn M)
    (hcard : 3 ≤ Fintype.card V) :
    ∃ N : SimpleGraph V,
      M ≤ N ∧ HasPlanarDrawingOn N ∧
        ∀ K : SimpleGraph V, N ≤ K → HasPlanarDrawingOn K → K = N := 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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