Planar edge upper bound from a drawing

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WilliamsonShmoys.planar_edge_upper_bound__p2m_53978d9a_ff21_4c38_b5ef_6e6d6a6a4bf9_a2

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

approximation-algorithmsplanar-graphstreewidth

A graph on at least three vertices with the supplied crossing-free drawing has at most 3 * |V| - 6 edges, written as edgeFinset.card + 6 ≤ 3 * |V|. The statement retains the target's exact drawing predicate and includes the three-vertex case.

Preamble

import Mathlib
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn

open WilliamsonShmoys
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_edge_upper_bound__p2m_53978d9a_ff21_4c38_b5ef_6e6d6a6a4bf9_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V)
    (hcard : LE.le 3 (Fintype.card V))
    (hKdraw : HasPlanarDrawingOn K) :
    letI : DecidableRel K.Adj := Classical.decRel K.Adj
    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).

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