Existence of an edge-maximal planar supergraph

Proved

WilliamsonShmoys.exists_maximal_planar_supergraph__p2m_d6b9112d_6e11_42e8_8b1b_3d46cf1b698c_a1

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

approximation-algorithmsplanar-graphstreewidth

Every drawable graph on the given finite vertex set has an inclusion-maximal drawable supergraph on that same vertex set. This isolates the finite extension construction from the edge-count argument. The vertex-count hypothesis is retained from the parent; the finite-maximization argument itself works for any finite vertex set.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.exists_maximal_planar_supergraph__p2m_d6b9112d_6e11_42e8_8b1b_3d46cf1b698c_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V) (hcard : 3 ≤ Fintype.card V)
    (hG : HasPlanarDrawingOn G) :
    ∃ K : SimpleGraph V, G ≤ K ∧ HasPlanarDrawingOn K ∧
      ∀ L : SimpleGraph V, K ≤ L → 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).

View graph