Disconnectedness yields an unreachable pair

Proved

WilliamsonShmoys.disconnected_has_unreachable_pair__p2m_834fbbd6_f11e_4933_84c3_a729cabab29e_a1

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

approximation-algorithmsplanar-graphstreewidth

A disconnected graph on a nonempty finite vertex type has two vertices that are not reachable from one another. The stated cardinality bound ensures nonemptiness.

Preamble

import Mathlib
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn

open WilliamsonShmoys
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.disconnected_has_unreachable_pair__p2m_834fbbd6_f11e_4933_84c3_a729cabab29e_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V)
    (hcard : 3 ≤ Fintype.card V)
    (hdis : ¬ H.Connected) :
    ∃ u v : V, ¬ H.Reachable u v := 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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