Finite maximal drawable extension

Proved

WilliamsonShmoys.planar_drawable_maximal_extension__p2m_bef9e736_66c4_4a44_9d5c_f3c0ae5dba68_a1

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

approximation-algorithmsplanar-graphstreewidth

Every drawable simple graph on a finite ordered vertex type has an inclusion-maximal drawable supergraph on the same vertices, with at least as many edges.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_drawable_maximal_extension__p2m_bef9e736_66c4_4a44_9d5c_f3c0ae5dba68_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (N : SimpleGraph V) (hN : HasPlanarDrawingOn N) :
    (letI : DecidableRel N.Adj := Classical.decRel N.Adj;
      ∃ H : SimpleGraph V,
        (letI : DecidableRel H.Adj := Classical.decRel H.Adj;
          N ≤ H ∧ HasPlanarDrawingOn H ∧
          N.edgeFinset.card ≤ H.edgeFinset.card ∧
          ∀ K : SimpleGraph V, H ≤ K → 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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