Finite maximal planar extension

Open

WilliamsonShmoys.maximal_planar_extension__p2m_ac4b6b7c_6a36_45ce_9f61_962873444f22_a2

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

approximation-algorithmsplanar-graphstreewidth

Every finite planar graph on at least three vertices is contained in a planar graph on the same vertices that is maximal under adding edges.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.maximal_planar_extension__p2m_ac4b6b7c_6a36_45ce_9f61_962873444f22_a2 {V : Type} [Fintype V] [LinearOrder V] (G : SimpleGraph V) (hG : WilliamsonShmoys.HasPlanarDrawingOn G) (hV : 3 <= Fintype.card V) :
    ∃ H : SimpleGraph V,
      (∀ u v, G.Adj u v → H.Adj u v) ∧
      WilliamsonShmoys.HasPlanarDrawingOn H ∧
      (∀ K : SimpleGraph V, (∀ u v, H.Adj u v → K.Adj u v) →
        WilliamsonShmoys.HasPlanarDrawingOn K → (∀ u v, K.Adj u v → H.Adj u v)) := 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