An edge-free cut separates its sides
ProvedWilliamsonShmoys.cut_implies_not_reachable__p2m_05c1a0b4_69ca_4d36_9f49_0774601d7f45_a3approximation-algorithmsplanar-graphstreewidth
If every edge crossing from a set of vertices to its complement is forbidden, then a vertex in the set cannot be reachable from a vertex outside it.
Preamble
import Mathlib
Formal statement
theorem WilliamsonShmoys.cut_implies_not_reachable__p2m_05c1a0b4_69ca_4d36_9f49_0774601d7f45_a3
{V : Type} [Fintype V] [LinearOrder V]
(G : SimpleGraph V) {u v : V} (A : Set V)
(huA : u ∈ A) (hvA : v ∉ A)
(hcut : ∀ {x y : V}, x ∈ A → y ∉ A → ¬ G.Adj x y) :
¬ 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).