A planar drawing can bridge distinct components

Open

WilliamsonShmoys.planar_drawing_admits_component_bridge__p2m_e7f58e98_9faf_49a6_bc58_50960cb4a0da_a1

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

approximation-algorithmsplanar-graphstreewidth

If two vertices are in distinct components of a graph with a crossing-free drawing, there is a planar drawing of a graph extending the original and containing the edge between those vertices. The core argument routes a simple connector through the unbounded face.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_drawing_admits_component_bridge__p2m_e7f58e98_9faf_49a6_bc58_50960cb4a0da_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V) (hG : WilliamsonShmoys.HasPlanarDrawingOn G)
    {u v : V} (hUnreachable : ¬ G.Reachable u v) :
    ∃ K : SimpleGraph V,
      G ≤ K ∧ K.Adj u v ∧ WilliamsonShmoys.HasPlanarDrawingOn K := 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