A drawable graph has a maximal drawable supergraph

Proved

WilliamsonShmoys.exists_maximal_planar_supergraph__p2m_4c570e3e_7abb_4bee_95ce_463a755b298b_a2

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 an inclusion-maximal drawable supergraph on that same vertex type.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.exists_maximal_planar_supergraph__p2m_4c570e3e_7abb_4bee_95ce_463a755b298b_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V) (hdraw : WilliamsonShmoys.HasPlanarDrawingOn G) :
    letI : ∀ G : SimpleGraph V, DecidableRel G.Adj := fun G => Classical.decRel G.Adj
    ∃ H : SimpleGraph V, G ≤ H ∧ WilliamsonShmoys.HasPlanarDrawingOn H ∧
      ∀ K : SimpleGraph V, H ≤ K → WilliamsonShmoys.HasPlanarDrawingOn K → K = 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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