Connected planar graph edge bound

Proved

WilliamsonShmoys.planar_connected_edge_bound__p2m_2e520467_0d96_4c71_b340_b60069871ecd_a1

by Eyal1990 · Oct 30, 2026 · Mathlib 0df444a (Lean v4.33.1)

approximation-algorithmsplanar-graphstreewidth

A finite simple connected graph with a crossing-free drawing and at least three vertices has at most three times its number of vertices minus six edges.

Preamble

import Mathlib
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_connected_edge_bound__p2m_2e520467_0d96_4c71_b340_b60069871ecd_a1 {V : Type} [Fintype V] [LinearOrder V] (H : SimpleGraph V) (hH : HasPlanarDrawingOn H) (hconn : H.Connected) (hV : 3 <= Fintype.card V) : H.edgeSet.ncard + 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