A disconnected planar drawing has a strict planar extension

Open

WilliamsonShmoys.disconnected_planar_graph_has_strict_planar_extension__p2m_2260a117_6439_491e_b3cc_351c9a5eff37_a1

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

approximation-algorithmsplanar-graphstreewidth

Given a disconnected simple graph on at least three vertices with a drawing satisfying HasPlanarDrawingOn, there is a strictly larger simple graph on the same vertices that also has such a drawing. The intended geometric argument joins exposed vertices from distinct components through the unbounded face.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.disconnected_planar_graph_has_strict_planar_extension__p2m_2260a117_6439_491e_b3cc_351c9a5eff37_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V) (hcard : 3 ≤ Fintype.card V)
    (hG : WilliamsonShmoys.HasPlanarDrawingOn G) (hDisconnected : ¬ G.Connected) :
    ∃ K : SimpleGraph V,
      G ≤ K ∧ WilliamsonShmoys.HasPlanarDrawingOn K ∧ K ≠ G := 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).

View graph