Reachability defines an edge-free cut

Proved

WilliamsonShmoys.reachable_cut__p2m_bb71a8e8_4f54_44cc_a603_6794a7a0cd2a_a2

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

approximation-algorithmsplanar-graphstreewidth

The vertices reachable from u form a set containing u and excluding v; no edge crosses from that set to its complement.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.reachable_cut__p2m_bb71a8e8_4f54_44cc_a603_6794a7a0cd2a_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V) {u v : V} (hdis : ¬ G.Reachable u v) :
    ∃ A : Set V, u ∈ A ∧ v ∉ A ∧
      ∀ ⦃x y : V⦄, x ∈ A → y ∉ A → ¬ G.Adj x y := 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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