Equivalence between pathbreadth and strong pathbreadth

We say that a given graph G = (V, E) has pathbreadth at most ρ, denoted pb(G) ≤ ρ, if there exists a Robertson and Seymour’s path decomposition where every bag is contained in the ρ-neighbourhood of some vertex. Similarly, we say that G has strong pathbreadth at most ρ, denoted spb(G) ≤ ρ, if there exists a Robertson and Seymour’s path decomposition where every bag is the complete ρ-neighbourhood of some vertex. It is straightforward that pb(G) ≤ spb(G) for any graph G. Inspired from a close conjecture in [Leitert and Dragan, COCOA’16], we prove in this note that spb(G) ≤ 4 · pb(G).