A maximal drawable simple graph has the triangulated edge count

Proved

WilliamsonShmoys.maximal_planar_edge_count__p2m_1634faf2_3284_496c_bfc9_f8ff236eda71_a1

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

approximation-algorithmsplanar-graphstreewidth

A drawable simple graph on at least three vertices that has no strictly larger drawable supergraph on the same vertices has exactly three times the number of vertices minus six edges.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.maximal_planar_edge_count__p2m_1634faf2_3284_496c_bfc9_f8ff236eda71_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V) (hG : HasPlanarDrawingOn G)
    (hV : 3 ≤ Fintype.card V)
    (hmax : ∀ M : SimpleGraph V, G ≤ M → HasPlanarDrawingOn M → M = G) :
    (letI : DecidableRel G.Adj := Classical.decRel G.Adj
     letI : Fintype ↑G.edgeSet := Fintype.ofFinite _
     3 * Fintype.card V = G.edgeFinset.card + 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).

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