Planar edge count upper bound on a fixed vertex set

Proved

WilliamsonShmoys.planar_edge_count_upper_bound_on

by Eyal1990 · Oct 26, 2026 · Mathlib 0df444a (Lean v4.33.1)

approximation-algorithmsplanar-graphstreewidth

Every graph with the specified planar drawing on at least three finite ordered vertices has at most three times its vertex count minus six edges. The boundary n = 3 is included; the inequality is valid even for disconnected graphs.

Preamble

import Mathlib
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_edge_count_upper_bound_on
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V)
    (hG : WilliamsonShmoys.HasPlanarDrawingOn G)
    (hV : 3 ≤ Fintype.card V) :
    G.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).

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