Reduce the planar edge bound to connected drawings

Proved

WilliamsonShmoys.planar_edge_count_from_connected__p2m_8a1edb89_91b7_4c19_a333_1779b45ebb8a_a1

by Eyal1990 · Oct 31, 2026 · Mathlib 0df444a (Lean v4.33.1)

approximation-algorithmsplanar-graphstreewidth

If every connected simple graph on this same finite ordered vertex type with a planar drawing and at least three vertices satisfies the edge bound, then every graph with a planar drawing on the type satisfies it.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_edge_count_from_connected__p2m_8a1edb89_91b7_4c19_a333_1779b45ebb8a_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V) (dG : DecidableRel G.Adj)
    (hconnBound : ∀ (H : SimpleGraph V) (dH : DecidableRel H.Adj),
      H.Connected → HasPlanarDrawingOn H → 3 ≤ Fintype.card V →
      (letI : DecidableRel H.Adj := dH
       H.edgeFinset.card ≤ 3 * Fintype.card V - 6))
    (hG : HasPlanarDrawingOn G)
    (hcard : 3 ≤ Fintype.card V) :
    (letI : DecidableRel G.Adj := dG
     G.edgeFinset.card ≤ 3 * Fintype.card V - 6) := 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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