An edge-maximal drawable graph has the triangulation edge count

Proved

WilliamsonShmoys.maximal_planar_drawing_edge_count__p2m_affb71eb_e232_4b0d_805e_31ff149cd3a9_a1

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

approximation-algorithmsplanar-graphstreewidth

A drawable simple graph on at least three vertices that has no strictly larger drawable supergraph on those vertices has exactly 3|V|−6 edges.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.maximal_planar_drawing_edge_count__p2m_affb71eb_e232_4b0d_805e_31ff149cd3a9_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (L : SimpleGraph V)
    (hcard : 3 ≤ Fintype.card V)
    (hL : WilliamsonShmoys.HasPlanarDrawingOn L)
    (hmax : ∀ M : SimpleGraph V,
      L.edgeSet ⊆ M.edgeSet →
      WilliamsonShmoys.HasPlanarDrawingOn M →
      M.edgeSet = L.edgeSet) :
    L.edgeSet.ncard = 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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