Edge bound for a finite simply drawn planar graph

Proved

WilliamsonShmoys.planar_edge_bound__p2m_278b0a4b_e4b3_4f5b_934c_d513e35b19a5_a1

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

approximation-algorithmsplanar-graphstreewidth

Every finite simple graph with a crossing-free drawing as defined by HasPlanarDrawingOn has at most three times its number of vertices minus three edges, with subtraction interpreted in Nat. This includes zero, one, and two vertices.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_edge_bound__p2m_278b0a4b_e4b3_4f5b_934c_d513e35b19a5_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V)
    (hG : HasPlanarDrawingOn G) :
    letI : DecidableRel G.Adj := Classical.decRel _
    G.edgeFinset.card ≤ 3 * Fintype.card V - 3 := 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).

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