Extend a connected planar graph to a maximal planar graph

Proved

WilliamsonShmoys_planar_maximal_extension__p2m_a74ab9cf_0d81_499a_8717_2a45ea195d9c_a1

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

approximation-algorithmsplanar-graphstreewidth

A connected finite planar graph has a connected planar supergraph on the same vertices that admits no proper planar supergraph on those vertices.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib
open WilliamsonShmoys
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys_planar_maximal_extension__p2m_a74ab9cf_0d81_499a_8717_2a45ea195d9c_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V) (hG : HasPlanarDrawingOn G) (hconn : G.Connected) :
    ∃ K : SimpleGraph V, G ≤ K ∧ HasPlanarDrawingOn K ∧ K.Connected ∧
      ∀ 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).

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