Disconnectedness yields a nonreachable pair
ProvedWilliamsonShmoys.disconnected_planar_drawing_has_nonreachable_pair__p2m_e7f58e98_9faf_49a6_bc58_50960cb4a0da_a1approximation-algorithmsplanar-graphstreewidth
A disconnected finite graph with at least three vertices has a pair of vertices in different connected components.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.disconnected_planar_drawing_has_nonreachable_pair__p2m_e7f58e98_9faf_49a6_bc58_50960cb4a0da_a1
{V : Type} [Fintype V] [LinearOrder V]
(G : SimpleGraph V) (hcard : 3 ≤ Fintype.card V)
(hDisconnected : ¬ G.Connected) :
∃ u v : V, ¬ G.Reachable u v := 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).