Edge count of a maximal drawable graph

Proved

WilliamsonShmoys.maximal_drawable_edge_count__p2m_ea8ef902_b5b0_4092_a824_9484bf1d76fc_a1

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

approximation-algorithmsplanar-graphstreewidth

An inclusion-maximal drawable simple graph on a finite linearly ordered vertex type with at least three vertices has exactly 3 * |V| - 6 edges. The proof route is to establish that maximality forces a triangulation, then combine Euler's formula with face-edge incidence counting; the three-vertex case is included.

Preamble

import Mathlib
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn

open WilliamsonShmoys
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.maximal_drawable_edge_count__p2m_ea8ef902_b5b0_4092_a824_9484bf1d76fc_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V)
    (hcard : LE.le 3 (Fintype.card V))
    (hKdraw : HasPlanarDrawingOn K)
    (hmax : forall L : SimpleGraph V, LE.le K L -> HasPlanarDrawingOn L -> L = K) :
    letI : DecidableRel K.Adj := Classical.decRel K.Adj
    3 * Fintype.card V = K.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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