Add an edge between distinct components of a planar drawing

Disproved

WilliamsonShmoys.planar_drawing_one_edge_augmentation__p2m_b6f7b032_7d17_4abc_a25e_6614208f3ad2_a1

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

approximation-algorithmsplanar-graphstreewidth

Every disconnected finite graph with a crossing-free drawing has two vertices in distinct components whose edge can be added while preserving a crossing-free drawing. The new graph has exactly the old edges plus that one undirected edge.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_drawing_one_edge_augmentation__p2m_b6f7b032_7d17_4abc_a25e_6614208f3ad2_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V)
    (hK : WilliamsonShmoys.HasPlanarDrawingOn K)
    (hnc : ¬ K.Connected) :
    ∃ a b : V, ∃ H : SimpleGraph V,
      ¬ K.Adj a b ∧ ¬ K.Reachable a b ∧
      (∀ u v, H.Adj u v ↔
        K.Adj u v ∨ (u = a ∧ v = b) ∨ (u = b ∧ v = a)) ∧
      WilliamsonShmoys.HasPlanarDrawingOn H := 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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