A disconnected graph has a strict edge extension from an unreachable pair

Proved

WilliamsonShmoys.disconnected_pair_edge_strict_extension__p2m_583909d6_9a92_4051_aa1a_1f45dfd3ea8e_a2

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

approximation-algorithmsplanar-graphstreewidth

For a finite ordered graph with at least three vertices that is disconnected, select vertices in different components. They are unreachable from one another, and adjoining the edge between them produces a strict supergraph.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.disconnected_pair_edge_strict_extension__p2m_583909d6_9a92_4051_aa1a_1f45dfd3ea8e_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V) (hDisconnected : ¬ H.Connected)
    (hn : Nat.le 3 (Fintype.card V)) :
    ∃ u v : V,
      ¬ H.Reachable u v ∧
      H ≤ H ⊔ SimpleGraph.edge u v ∧
      H ≠ H ⊔ SimpleGraph.edge 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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