Maximal drawable extension is connected

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WilliamsonShmoys.maximal_planar_extension_connected__p2m_c5c73f75_8c1e_44ef_9deb_26150704a428_a1

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

approximation-algorithmsplanar-graphstreewidth

A drawable graph on at least three vertices that has no proper drawable supergraph on the same vertices is connected.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.maximal_planar_extension_connected__p2m_c5c73f75_8c1e_44ef_9deb_26150704a428_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V) (hcard : 3 <= Fintype.card V)
    (hH : WilliamsonShmoys.HasPlanarDrawingOn H)
    (hmax : forall J : SimpleGraph V, H <= J ->
      WilliamsonShmoys.HasPlanarDrawingOn J -> H = J) :
    H.Connected := 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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