Edge count of a maximal planar graph

Proved

WilliamsonShmoys.maximal_planar_edge_bound__p2m_58a4bbec_9b1c_4062_8fb0_876ba124b0f6_a1

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

approximation-algorithmsplanar-graphstreewidth

A finite simple graph on at least three vertices that is maximal among planar graphs on that vertex set has exactly 3|V| − 6 edges.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.maximal_planar_edge_bound__p2m_58a4bbec_9b1c_4062_8fb0_876ba124b0f6_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V) [DecidableRel H.Adj]
    (hH : HasPlanarDrawingOn H)
    (hmax : ∀ K : SimpleGraph V,
      (∀ u v : V, H.Adj u v → K.Adj u v) →
      HasPlanarDrawingOn K → K = H)
    (hcard : 3 ≤ Fintype.card V) :
    H.edgeFinset.card + 6 = 3 * Fintype.card 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).

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