Edge count of a maximal drawable graph

Proved

WilliamsonShmoys.maximal_planar_drawingOn_edge_count__p2m_c96f9502_b6a7_485d_895c_b4be9d2d28fa_a2

by Eyal1990 · Oct 31, 2026 · Mathlib 0df444a (Lean v4.33.1)

approximation-algorithmsplanar-graphstreewidth

A drawable graph on at least three finite linearly ordered vertices with no strictly larger drawable supergraph has exactly 3|V| − 6 edges.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.maximal_planar_drawingOn_edge_count__p2m_c96f9502_b6a7_485d_895c_b4be9d2d28fa_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V)
    (hK : WilliamsonShmoys.HasPlanarDrawingOn K)
    (hmax : ∀ L : SimpleGraph V, K ≤ L →
      WilliamsonShmoys.HasPlanarDrawingOn L → L ≤ K)
    (hcard : 3 ≤ Fintype.card V) :
    letI : DecidableRel K.Adj := Classical.decRel K.Adj
    K.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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