Existence of a maximal drawable supergraph

Proved

WilliamsonShmoys.maximal_planar_extension__p2m_9ca2ebd3_f99a_4019_8342_a41927f598bc_a3

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

approximation-algorithmsplanar-graphstreewidth

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

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.maximal_planar_extension__p2m_9ca2ebd3_f99a_4019_8342_a41927f598bc_a3
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V) [DecidableRel H.Adj]
    (hH : HasPlanarDrawingOn H) :
    ∃ (K : SimpleGraph V) (dK : DecidableRel K.Adj),
      (letI : DecidableRel K.Adj := dK
       HasPlanarDrawingOn K ∧
       (∀ (L : SimpleGraph V) [DecidableRel L.Adj],
         HasPlanarDrawingOn L →
         K.edgeFinset ⊆ L.edgeFinset →
         K.edgeFinset = L.edgeFinset) ∧
       H.edgeFinset ⊆ K.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).

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