Maximal planar drawing edge count lower bound

Proved

maximal_planar_drawing_edge_count_lower__p2m_f064fab3_2a2a_43e1_9be9_e5e958f9fe83_a1

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

approximation-algorithmsplanar-graphstreewidth

A graph with a crossing-free drawing that is maximal among drawable supergraphs on the same vertex type, with at least three vertices, has at least 3|V|−6 edges.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem maximal_planar_drawing_edge_count_lower__p2m_f064fab3_2a2a_43e1_9be9_e5e958f9fe83_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V) [DecidableRel K.Adj]
    (hK : HasPlanarDrawingOn K)
    (hmax : ∀ L : SimpleGraph V, K ≤ L → HasPlanarDrawingOn L → L = K)
    (hcard : 3 ≤ Fintype.card V) :
    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).

View graph