Disconnectedness yields an unreachable pair

Proved

WilliamsonShmoys.disconnected_has_unreachable_pair__p2m_37fa9a5a_5453_43d3_9231_e1df50e08028_a1

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

approximation-algorithmsplanar-graphstreewidth

For a finite vertex type with at least three elements, disconnectedness supplies two vertices that are not connected by a path. This separates the graph-theoretic witness extraction from the geometric edge-addition argument.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.disconnected_has_unreachable_pair__p2m_37fa9a5a_5453_43d3_9231_e1df50e08028_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V)
    (hcard : 3 <= Fintype.card V)
    (hDisconnected : ¬ K.Connected) :
    ∃ u v : V, ¬ K.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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