Reduce the planar edge bound to connected graphs

Proved

WilliamsonShmoys.planar_drawing_edge_upper_bound_component_reduction__p2m_263e0430_0bed_4695_a55f_33a5be744a5b_a1

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

approximation-algorithmsplanar-graphstreewidth

If every connected finite graph with a drawing satisfying HasPlanarDrawingOn and at least three vertices obeys the planar edge bound, then every such finite graph with at least three vertices obeys it, whether connected or not. The reduction must account separately for one-vertex and two-vertex components.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_drawing_edge_upper_bound_component_reduction__p2m_263e0430_0bed_4695_a55f_33a5be744a5b_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (N : SimpleGraph V) (hV : 3 <= Fintype.card V)
    (hN : HasPlanarDrawingOn N)
    (hconnected : ∀ {W : Type} [Fintype W] [LinearOrder W]
      (G : SimpleGraph W) (hW : 3 <= Fintype.card W)
      (hconn : G.Connected) (hG : HasPlanarDrawingOn G),
      (letI : DecidableRel G.Adj := Classical.decRel G.Adj;
       G.edgeFinset.card + 6 <= 3 * Fintype.card W)) :
    (letI : DecidableRel N.Adj := Classical.decRel N.Adj;
     N.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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