Attach a planar edge across components with a size bound

Proved

WilliamsonShmoys.planar_nonconnected_edge_attachment__p2m_7d9aa82d_f2fe_4aa2_b79e_d2d48fb9feb3_a5

by Eyal1990 · Oct 31, 2026 · Mathlib 0df444a (Lean v4.33.1)

approximation-algorithmsplanar-graphstreewidth

For a disconnected finite graph on at least three vertices with a planar drawing, add exactly one edge between distinct unreachable vertices and retain a planar drawing.

Preamble

import Mathlib
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn

set_option autoImplicit false
open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_nonconnected_edge_attachment__p2m_7d9aa82d_f2fe_4aa2_b79e_d2d48fb9feb3_a5
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V) (hG : WilliamsonShmoys.HasPlanarDrawingOn G)
    (hcard : 3 <= Fintype.card V) (hdis : ¬ G.Connected) :
    ∃ (u v : V) (huv : u ≠ v),
      ¬ G.Reachable u v ∧
      ∃ H : SimpleGraph V,
        G ≤ H ∧ WilliamsonShmoys.HasPlanarDrawingOn H ∧
        ∀ a b, H.Adj a b ↔
          G.Adj a b ∨ (a = u ∧ b = v) ∨ (a = v ∧ b = u) := 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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