Extend a planar drawing by a bridge between components

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WilliamsonShmoys.planar_drawing_add_bridge_geometry__p2m_68fcfb6b_9fe7_4d9c_85f0_a7b7f89a4ef9_a2

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

approximation-algorithmsplanar-graphstreewidth

Given a planar drawing and two nonadjacent vertices in different components, add exactly their undirected edge and produce a planar drawing of the resulting graph.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_drawing_add_bridge_geometry__p2m_68fcfb6b_9fe7_4d9c_85f0_a7b7f89a4ef9_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V)
    (hK : WilliamsonShmoys.HasPlanarDrawingOn K)
    (a b : V)
    (hab : ¬ K.Adj a b)
    (hnotReachable : ¬ K.Reachable a b) :
    ∃ H : SimpleGraph V,
      (∀ u v, H.Adj u v ↔
        K.Adj u v ∨ (u = a ∧ v = b) ∨ (u = b ∧ v = a)) ∧
      WilliamsonShmoys.HasPlanarDrawingOn H := 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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