Unreachable vertices determine an edge-free component cut

Proved

WilliamsonShmoys.planar_unreachable_vertices_component_cut__p2m_95b2a4a5_f4af_45e5_83a3_0810e97a8978_a2

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

approximation-algorithmsplanar-graphstreewidth

If a cannot reach b, there is a set of vertices containing a but not b such that every edge has both endpoints in the set or both outside it.

Preamble

import Mathlib
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_unreachable_vertices_component_cut__p2m_95b2a4a5_f4af_45e5_83a3_0810e97a8978_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V) (a b : V) (hnotReach : ¬ K.Reachable a b) :
    ∃ P : Set V, P a ∧ ¬ P b ∧
      ∀ u v, K.Adj u v → (P u ↔ P 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).

View graph