Drawable strict extension of a disconnected graph

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WilliamsonShmoys.planar_extension_of_disconnected_graph__p2m_8a39e0cf_ef07_4e8d_9689_6d4757ac8893_a1

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

approximation-algorithmsplanar-graphstreewidth

Every disconnected drawable graph on a finite linearly ordered vertex type with at least three vertices has a drawable strict supergraph on that same vertex type.

Preamble

import Mathlib
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_extension_of_disconnected_graph__p2m_8a39e0cf_ef07_4e8d_9689_6d4757ac8893_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V) (hcard : 3 ≤ Fintype.card V)
    (hK : HasPlanarDrawingOn K) (hdis : ¬ K.Connected) :
    ∃ L : SimpleGraph V, K ≤ L ∧ HasPlanarDrawingOn L ∧ ¬ L ≤ K := 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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