Add a planar edge between disconnected vertices

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WilliamsonShmoys.planar_drawing_one_step_extension__p2m_e072abf9_a18d_4ec4_b40d_84b33bde54a7_a1

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

approximation-algorithmsplanar-graphstreewidth

For a finite linearly ordered vertex type, if a graph has a planar drawing and two vertices are in different connected components, there is a planar supergraph on the same vertices that retains every old edge and adds an edge between those vertices.

Preamble

import Mathlib
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_drawing_one_step_extension__p2m_e072abf9_a18d_4ec4_b40d_84b33bde54a7_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V)
    (hG : HasPlanarDrawingOn G)
    (u v : V)
    (hsep : ¬ G.Reachable u v) :
    ∃ K : SimpleGraph V,
      HasPlanarDrawingOn K ∧ G ≤ K ∧ K.Adj u 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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