Maximal drawable graphs have the triangulation edge count

Proved

WilliamsonShmoys.maximal_drawable_has_exact_edge_count__p2m_611c5e27_b46d_4fae_a676_63177eb6dd3b_a3

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

approximation-algorithmsplanar-graphstreewidth

A drawable graph on a finite linearly ordered vertex type with at least three vertices has exactly three times the vertex count minus six edges if every drawable supergraph on that type is contained in it.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false
open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.maximal_drawable_has_exact_edge_count__p2m_611c5e27_b46d_4fae_a676_63177eb6dd3b_a3
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V) (hV : 3 ≤ Fintype.card V)
    (hPlanar : HasPlanarDrawingOn K)
    (hMaximal : ∀ H : SimpleGraph V, K ≤ H → HasPlanarDrawingOn H → H ≤ K) :
    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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