Add a bridge between unreachable vertices in a planar drawing

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WilliamsonShmoys.planar_drawing_add_bridge_between_unreachable_vertices__p2m_834fbbd6_f11e_4933_84c3_a729cabab29e_a1

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

approximation-algorithmsplanar-graphstreewidth

Given a planar drawing and two unreachable vertices, adding the edge between them preserves the existence of a planar drawing on the same vertex set.

Preamble

import Mathlib
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn

open WilliamsonShmoys
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_drawing_add_bridge_between_unreachable_vertices__p2m_834fbbd6_f11e_4933_84c3_a729cabab29e_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V)
    (hH : HasPlanarDrawingOn H)
    (u v : V)
    (hnot : ¬ 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).

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