Maximality forces the triangulation edge lower bound

Open

WilliamsonShmoys.maximal_planar_drawing_edge_count_lower_bound__p2m_88aeae11_6627_4561_98a0_139448671fc8_a1

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

approximation-algorithmsplanar-graphstreewidth

An inclusion-maximal graph with the specified crossing-free drawing and at least three vertices has at least 3|V|−6 edges.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.maximal_planar_drawing_edge_count_lower_bound__p2m_88aeae11_6627_4561_98a0_139448671fc8_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V) (dG : DecidableRel G.Adj)
    (hG : HasPlanarDrawingOn G)
    (hn : 3 ≤ Fintype.card V)
    (hmax : ∀ (K : SimpleGraph V) (dK : DecidableRel K.Adj),
      letI : DecidableRel G.Adj := dG
      letI : DecidableRel K.Adj := dK
      HasPlanarDrawingOn K →
      G.edgeFinset ⊆ K.edgeFinset →
      K.edgeFinset = G.edgeFinset) :
    letI : DecidableRel G.Adj := dG
    3 * Fintype.card V - 6 ≤ G.edgeFinset.card := 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