A drawable graph has a maximum drawable supergraph

Proved

WilliamsonShmoys.exists_maximal_drawable_extension__p2m_067ebb73_e756_4640_b5cf_ef3a010c6f28_a1

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

approximation-algorithmsplanar-graphstreewidth

Every drawable graph on a finite vertex type has a drawable supergraph on that same type whose edge count is at least as large and is maximum among drawable supergraphs. This removes the finite choice and maximality setup from the parent. Independently, enumerate the finitely many simple graphs on V, restrict to drawable supergraphs of Q, and choose one with maximum edge count; Q ensures the candidate set is nonempty.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.exists_maximal_drawable_extension__p2m_067ebb73_e756_4640_b5cf_ef3a010c6f28_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (Q : SimpleGraph V) (hQ : HasPlanarDrawingOn Q) :
    ∃ R : SimpleGraph V,
      (∀ u v, Q.Adj u v → R.Adj u v) ∧
      HasPlanarDrawingOn R ∧
      (letI : DecidableRel Q.Adj := Classical.decRel Q.Adj
       letI : DecidableRel R.Adj := Classical.decRel R.Adj
       Q.edgeFinset.card ≤ R.edgeFinset.card) ∧
      (∀ S : SimpleGraph V, (∀ u v, R.Adj u v → S.Adj u v) →
        HasPlanarDrawingOn S →
        (letI : DecidableRel S.Adj := Classical.decRel S.Adj
         letI : DecidableRel R.Adj := Classical.decRel R.Adj
         S.edgeFinset.card ≤ R.edgeFinset.card)) := 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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