Add a planar bridge across an edge-free cut

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WilliamsonShmoys.planar_add_bridge_across_cut__p2m_bb71a8e8_4f54_44cc_a603_6794a7a0cd2a_a2

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

approximation-algorithmsplanar-graphstreewidth

A planar graph with an edge-free cut separating u and v has a planar supergraph on the same vertices that includes the edge u-v.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_add_bridge_across_cut__p2m_bb71a8e8_4f54_44cc_a603_6794a7a0cd2a_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V) (hG : HasPlanarDrawingOn G)
    {u v : V} (A : Set V) (huA : u ∈ A) (hvA : v ∉ A)
    (hcut : ∀ ⦃x y : V⦄, x ∈ A → y ∉ A → ¬ G.Adj x y) :
    ∃ K : SimpleGraph V,
      G ≤ K ∧ HasPlanarDrawingOn K ∧ K.Adj u 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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