Add a drawable bridge between two components

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WilliamsonShmoys.planar_drawing_add_reachable_pair__p2m_02da0ff6_1020_4ffd_8f79_c5b3eb2fc9a4_a2

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

approximation-algorithmsplanar-graphstreewidth

Given a finite drawing and two vertices in different components, add edges to obtain a supergraph that still has a drawing and in which those vertices are connected by a path. A proof can find a face incident to two components and draw a simple joining arc inside that face; the central work is formalizing the face and arc construction while preserving all existing arcs.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_drawing_add_reachable_pair__p2m_02da0ff6_1020_4ffd_8f79_c5b3eb2fc9a4_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V) [DecidableRel H.Adj]
    (hH : HasPlanarDrawingOn H) {a b : V}
    (hab : ¬ H.Reachable a b) :
    ∃ G : SimpleGraph V, H ≤ G ∧ HasPlanarDrawingOn G ∧ G.Reachable a b := 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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