Add an edge between suitable components of a disconnected planar drawing

Disproved

WilliamsonShmoys.planar_drawing_disconnected_extension__p2m_29b7bd89_c713_464d_96d1_5b54a307b779_a1

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

approximation-algorithmsplanar-graphstreewidth

For a finite graph with the given crossing-free drawing, if it is disconnected, there are unreachable vertices whose connecting edge can be added while retaining a crossing-free drawing.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib
open WilliamsonShmoys
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_drawing_disconnected_extension__p2m_29b7bd89_c713_464d_96d1_5b54a307b779_a1 {V : Type} [Fintype V] [LinearOrder V] (H : SimpleGraph V) (hH : HasPlanarDrawingOn H) : ¬ H.Connected → ∃ u v, ¬ H.Reachable u v ∧ HasPlanarDrawingOn (H ⊔ SimpleGraph.fromEdgeSet ({s(u, v)} : Set (Sym2 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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