Extend a planar drawing to an inclusion-maximal planar supergraph

Proved

WilliamsonShmoys.planar_drawing_has_edge_maximal_extension__p2m_8f423dbe_14e3_49a1_828e_7cd49c2ddeb4_a1

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

approximation-algorithmsplanar-graphstreewidth

Every finite drawable graph on at least three vertices is contained in a drawable graph on the same vertices that has no strictly larger drawable simple supergraph.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_drawing_has_edge_maximal_extension__p2m_8f423dbe_14e3_49a1_828e_7cd49c2ddeb4_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V) (hcard : 3 ≤ Fintype.card V)
    (hdrawing : WilliamsonShmoys.HasPlanarDrawingOn H) :
    letI : ∀ K : SimpleGraph V, DecidableRel K.Adj := fun K => Classical.decRel K.Adj
    ∃ K : SimpleGraph V, H ≤ K ∧ WilliamsonShmoys.HasPlanarDrawingOn K ∧
      ∀ L : SimpleGraph V, K ≤ L → WilliamsonShmoys.HasPlanarDrawingOn L → L = K := 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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