Maximal drawable graphs have the triangulation edge count

Proved

WilliamsonShmoys.maximal_planar_edge_count__p2m_5e34af62_bc93_4e27_8f77_f0a9265acaf1_a1

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

approximation-algorithmsplanar-graphstreewidth

A drawable simple graph on at least three finite vertices that has no strictly larger drawable supergraph on those vertices has exactly $3|V|-6$ edges.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn

Formal statement

theorem WilliamsonShmoys.maximal_planar_edge_count__p2m_5e34af62_bc93_4e27_8f77_f0a9265acaf1_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V) (hcard : 3 ≤ Fintype.card V)
    (hK : WilliamsonShmoys.HasPlanarDrawingOn K)
    (hmax : ∀ H : SimpleGraph V, K ≤ H → WilliamsonShmoys.HasPlanarDrawingOn H → H ≤ K) :
    K.edgeSet.ncard = 3 * Fintype.card V - 6 := 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