Extend a planar drawing by an edge between unreachable vertices

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WilliamsonShmoys.planar_drawing_add_edge_between_separate_components__p2m_8e313f52_7cd9_446d_9cbb_05951652d57f_a3

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

approximation-algorithmsplanar-graphstreewidth

For a finite ordered vertex type, a graph with a planar drawing, and two unreachable vertices, there is a graph extension containing their edge that still has a planar drawing. Unreachability rules out the equality boundary u = v. This isolates the geometric re-embedding construction; the parent derives strictness from the added adjacency.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_drawing_add_edge_between_separate_components__p2m_8e313f52_7cd9_446d_9cbb_05951652d57f_a3
    {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).

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