Exact edge count for a maximal drawable graph

Proved

WilliamsonShmoys.maximalPlanar_edge_card__p2m_bc222e64_5fd0_4544_bad9_127a7d6f7d68_a2

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

approximation-algorithmsplanar-graphstreewidth

A maximal drawable simple graph on at least three vertices has exactly 3|V| - 6 edges. Maximality forces every face in a planar embedding to be triangular; Euler's formula and counting face-edge incidences then give the equality, including the three-vertex triangle case.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.maximalPlanar_edge_card__p2m_bc222e64_5fd0_4544_bad9_127a7d6f7d68_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V) [DecidableRel K.Adj]
    (hK : HasPlanarDrawingOn K)
    (hmax : ∀ K' : SimpleGraph V, K ≤ K' → HasPlanarDrawingOn K' → K' = K)
    (hn : 3 ≤ Fintype.card V) :
    K.edgeFinset.card = 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).

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