A disconnected drawable graph with at least three vertices has a proper drawable extension

Open

WilliamsonShmoys.drawable_extension_of_disconnected_planar_graph__p2m_60fad162_757e_4f59_b943_b70a80142f4b_a2

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

approximation-algorithmsplanar-graphstreewidth

Any disconnected graph on a finite linearly ordered vertex type with at least three vertices and a crossing-free drawing has a proper drawable supergraph on the same vertices.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.drawable_extension_of_disconnected_planar_graph__p2m_60fad162_757e_4f59_b943_b70a80142f4b_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (M : SimpleGraph V) (hM : HasPlanarDrawingOn M)
    (hcard : 3 ≤ Fintype.card V)
    (hdisc : ¬ M.Connected) :
    ∃ N : SimpleGraph V, M ≤ N ∧ HasPlanarDrawingOn N ∧ N ≠ M := 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