Select nonadjacent vertices in different components
DisprovedWilliamsonShmoys.planar_drawing_distinct_components_endpoints__p2m_68fcfb6b_9fe7_4d9c_85f0_a7b7f89a4ef9_a2approximation-algorithmsplanar-graphstreewidth
A graph that is not connected has two vertices that are not adjacent and are not reachable from each other.
Preamble
import Mathlib
Formal statement
theorem WilliamsonShmoys.planar_drawing_distinct_components_endpoints__p2m_68fcfb6b_9fe7_4d9c_85f0_a7b7f89a4ef9_a2
{V : Type} [Fintype V] [LinearOrder V]
(K : SimpleGraph V)
(hnc : ¬ K.Connected) :
∃ a b : V, ¬ K.Adj a b ∧ ¬ K.Reachable a b := by sorrySource
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).