Planar extension across distinct components

Proved

WilliamsonShmoys.planar_add_edge_between_unreachable_vertices__p2m_b0b687f8_4cc8_496c_af1a_60542cca02f0_a1

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

approximation-algorithmsplanar-graphstreewidth

For a graph with a crossing-free drawing, if two vertices lie in different connected components, there is a strictly larger graph on the same vertices that adds an edge between them and still has a crossing-free drawing.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_add_edge_between_unreachable_vertices__p2m_b0b687f8_4cc8_496c_af1a_60542cca02f0_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V) (hG : WilliamsonShmoys.HasPlanarDrawingOn G)
    (u v : V) (hUV : ¬ G.Reachable u v) :
    ∃ K : SimpleGraph V,
      G ≤ K ∧ K ≠ G ∧ 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).

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