A short note on the complexity of computing strong pathbreadth

The strong pathbreadth of a given graph G is the minimum ρ such that G admits a Robertson and Seymour's path decomposition where every bag is the complete ρ-neighbourhood of some vertex in G. We prove that deciding whether a given graph has strong pathbreadth at most one is NP-complete. The latter answers negatively to a conjecture of [Leitert and Dragan, CO-COA'16].