Simple planar edge bound

Proved

WilliamsonShmoys.simple_planar_edge_bound__p2m_b664cddc_4710_40ab_ab7d_a96b7cb12f1c_a2

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

approximation-algorithmsplanar-graphstreewidth

A finite connected simple graph with the specified crossing-free drawing and at least three vertices has at most 3n − 6 edges. The vertex lower bound is necessary for this formulation.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.simple_planar_edge_bound__p2m_b664cddc_4710_40ab_ab7d_a96b7cb12f1c_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V) [DecidableRel K.Adj]
    (hV : 3 ≤ Fintype.card V)
    (hconn : K.Connected)
    (hK : WilliamsonShmoys.HasPlanarDrawingOn K) :
    K.edgeFinset.card + 6 ≤ 3 * Fintype.card V := 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