Iterate planar bridges to connect a finite graph

Proved

WilliamsonShmoys.finite_planar_bridge_iteration__p2m_6542605d_05b7_4349_89df_2d6e0eb6b2d4_a1

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

approximation-algorithmsplanar-graphstreewidth

For a finite vertex type, if every disconnected planar supergraph of G can be extended by a planar bridge between distinct components, then G has a connected planar supergraph. Prove this by finite induction on the number of connected components; each bridge merges at least two components.

Preamble

import Mathlib
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.finite_planar_bridge_iteration__p2m_6542605d_05b7_4349_89df_2d6e0eb6b2d4_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V)
    (hplanar : HasPlanarDrawingOn G)
    (hcard : 3 <= Fintype.card V)
    (hstep : ∀ (K : SimpleGraph V),
      G.edgeSet ⊆ K.edgeSet →
      HasPlanarDrawingOn K →
      ¬ K.Connected →
      ∃ L : SimpleGraph V,
        K.edgeSet ⊆ L.edgeSet ∧
        (∃ u v, ¬ K.Reachable u v ∧ L.Adj u v) ∧
        HasPlanarDrawingOn L) :
    ∃ H : SimpleGraph V,
      G.edgeSet ⊆ H.edgeSet ∧ 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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