Upper bound on drawable planar graph edges

Proved

WilliamsonShmoys.planar_edge_count_upper_bound__p2m_022fc3fe_2112_4838_9b2a_056c25ecda08_a2

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

approximation-algorithmsplanar-graphstreewidth

Every drawable simple graph on at least three finite vertices has at most three times the number of vertices minus six edges.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_edge_count_upper_bound__p2m_022fc3fe_2112_4838_9b2a_056c25ecda08_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V) (dG : DecidableRel G.Adj)
    (hcard : 3 ≤ Fintype.card V)
    (hG : WilliamsonShmoys.HasPlanarDrawingOn G) :
    G.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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