Add an edge between components of a sufficiently large disconnected planar drawing

Open

WilliamsonShmoys.planar_drawing_disconnected_extension__p2m_29b7bd89_c713_464d_96d1_5b54a307b779_a3

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

approximation-algorithmsplanar-graphstreewidth

If a drawable graph on at least three vertices is disconnected, some two unreachable vertices can be joined by a new edge while retaining a planar 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_a3 {V : Type} [Fintype V] [LinearOrder V] (H : SimpleGraph V) (hH : HasPlanarDrawingOn H) (hcard : 3 ≤ Fintype.card V) : ¬ 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).

View graph