A maximal planar drawing has at least the triangulation edge count

Proved

WilliamsonShmoys.maximal_planar_drawing_edge_lower_bound__p2m_757b64a4_4ecc_451f_8c8e_f98c58cc600e_a1

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

approximation-algorithmsplanar-graphstreewidth

Every inclusion-maximal graph with a crossing-free drawing on at least three vertices has at least 3|V|−6 edges.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.maximal_planar_drawing_edge_lower_bound__p2m_757b64a4_4ecc_451f_8c8e_f98c58cc600e_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (L : SimpleGraph V) (hV : 3 ≤ Fintype.card V)
    (hL : HasPlanarDrawingOn L)
    (hMax : ∀ M : SimpleGraph V, L ≤ M → HasPlanarDrawingOn M → M = L) :
    (letI : DecidableRel L.Adj := Classical.decRel L.Adj;
      3 * Fintype.card V ≤ L.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).

View graph