Iterate planar bridge extensions to connect a drawing

Proved

WilliamsonShmoys.planar_bridge_steps_connect__p2m_50046e68_ab20_4f6d_a7b5_2cf104147317_a2

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

approximation-algorithmsplanar-graphstreewidth

If every disconnected graph on the fixed finite vertex type with a crossing-free drawing admits a new planar bridge extension, then any such drawing can be extended to a connected planar supergraph. Repeatedly adding a previously absent edge strictly increases the finite edge set, so the augmentation terminates.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_bridge_steps_connect__p2m_50046e68_ab20_4f6d_a7b5_2cf104147317_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V) (hK : HasPlanarDrawingOn K)
    (hstep : ∀ G : SimpleGraph V, HasPlanarDrawingOn G → ¬ G.Connected →
      ∃ u v : V, ∃ H : SimpleGraph V,
        ¬ G.Adj u v ∧ G ≤ H ∧ H.Adj u v ∧ HasPlanarDrawingOn H) :
    ∃ H : SimpleGraph V, K ≤ H ∧ H.Connected ∧ 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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