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