Existence of a maximal drawable supergraph

Proved

WilliamsonShmoys.planar_drawing_maximal_supergraph__p2m_49e59269_0942_49a7_bb42_8d7dcf799486_a1

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

approximation-algorithmsplanar-graphstreewidth

Every drawable finite simple graph has a drawable supergraph on the same vertices that is maximal under edge inclusion.

Preamble

import Mathlib
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_drawing_maximal_supergraph__p2m_49e59269_0942_49a7_bb42_8d7dcf799486_a1
    {U : Type} [Fintype U] [LinearOrder U]
    (H : SimpleGraph U)
    (hdraw : WilliamsonShmoys.HasPlanarDrawingOn H) :
    ∃ K : SimpleGraph U, H ≤ K ∧ WilliamsonShmoys.HasPlanarDrawingOn K ∧
      ∀ L : SimpleGraph U, K ≤ L → WilliamsonShmoys.HasPlanarDrawingOn L → K = 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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