Connected planar extension with at least three vertices

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WilliamsonShmoys.planar_drawing_connected_extension__p2m_2868a47b_b066_46c0_a57b_16dc6ed3b9ef_a3

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

approximation-algorithmsplanar-graphstreewidth

A disconnected finite graph on at least three vertices with a crossing-free drawing extends, on the same vertex set, to a connected simple graph with a crossing-free drawing, retaining every original edge.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib
open WilliamsonShmoys
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_drawing_connected_extension__p2m_2868a47b_b066_46c0_a57b_16dc6ed3b9ef_a3 {V : Type} [Fintype V] [LinearOrder V] (G : SimpleGraph V) [DecidableRel G.Adj] (hG : HasPlanarDrawingOn G) (hcard : 3 ≤ Fintype.card V) (hconn : ¬ G.Connected) : ∃ H : SimpleGraph V, G ≤ 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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