A disconnected graph has an unreachable nonadjacent pair

Proved

WilliamsonShmoys.planar_disconnected_has_separated_pair__p2m_46cfe7ce_b627_4e5a_9d08_1eab5cbb77ec_a2

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

approximation-algorithmsplanar-graphstreewidth

If a finite graph with at least three vertices is disconnected, it has two vertices that are neither reachable from one another nor adjacent.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_disconnected_has_separated_pair__p2m_46cfe7ce_b627_4e5a_9d08_1eab5cbb77ec_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (M : SimpleGraph V)
    (hn : 3 ≤ Fintype.card V)
    (hdis : ¬ M.Connected) :
    ∃ u v : V, ¬ M.Reachable u v ∧ ¬ M.Adj 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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