Maximal planar edge lower bound

Proved

WilliamsonShmoys.planar_maximal_edge_lower_bound__p2m_6582e0a3_76e8_4a07_8065_8715ca29a41a_a1

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

approximation-algorithmsplanar-graphstreewidth

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

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_maximal_edge_lower_bound__p2m_6582e0a3_76e8_4a07_8065_8715ca29a41a_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (N : SimpleGraph V) (hN : WilliamsonShmoys.HasPlanarDrawingOn N)
    (hcard : 3 ≤ Fintype.card V)
    (hmax : ∀ P : SimpleGraph V, N ≤ P → WilliamsonShmoys.HasPlanarDrawingOn P → P = N) :
    (letI : DecidableRel N.Adj := Classical.decRel N.Adj
     3 * Fintype.card V ≤ N.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