Existence of an edge-maximal drawable supergraph

Proved

WilliamsonShmoys.planar_maximal_supergraph__p2m_07b4009a_04ca_4342_a9d5_b274d08c0821_a1

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

approximation-algorithmsplanar-graphstreewidth

Every drawable finite simple graph has a drawable edge-maximal supergraph on the same vertex type. This is the finite maximality step: among the finitely many edge sets of drawable supergraphs, select one with maximum cardinality.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_maximal_supergraph__p2m_07b4009a_04ca_4342_a9d5_b274d08c0821_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V) (hG : HasPlanarDrawingOn G) :
    letI : ∀ H : SimpleGraph V, DecidableRel H.Adj :=
      fun H => Classical.decRel H.Adj
    ∃ H : SimpleGraph V,
      G.edgeFinset ⊆ H.edgeFinset ∧
      HasPlanarDrawingOn H ∧
      ∀ K : SimpleGraph V,
        H.edgeFinset ⊆ K.edgeFinset →
        HasPlanarDrawingOn K →
        K.edgeFinset ⊆ H.edgeFinset := 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).

View graph