A disconnected planar drawing admits an edge extension

Proved

WilliamsonShmoys.planar_drawing_disconnected_extension__p2m_46bd3770_49b2_4c1a_90e5_69dd719cc31d_a1

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

approximation-algorithmsplanar-graphstreewidth

For a drawable graph on at least three finite linearly ordered vertices, if the graph is disconnected, there are distinct nonadjacent vertices that can be joined by an edge so that the resulting supergraph is still drawable.

Preamble

import Mathlib
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_drawing_disconnected_extension__p2m_46bd3770_49b2_4c1a_90e5_69dd719cc31d_a1 {V : Type} [Fintype V] [LinearOrder V] (K : SimpleGraph V) [DecidableRel K.Adj] (hcard : 3 <= Fintype.card V) (hK : WilliamsonShmoys.HasPlanarDrawingOn K) (hconn : ¬ K.Connected) : ∃ x y : V, ∃ L : SimpleGraph V, x ≠ y ∧ ¬ K.Adj x y ∧ K ≤ L ∧ L.Adj x y ∧ WilliamsonShmoys.HasPlanarDrawingOn L := 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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