Equivalence between pathbreadth and strong pathbreadth

Guillaume Ducoffe (1, 2, 3) , Arne Leitert (4)

1 ICUB - Research Institute of the University of Bucharest
2 UniBuc - University of Bucharest
3 ICI - National Institute for Research and Development in Informatics [Bucharest]
4 Central Washington University

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).

Mots clés: strong pathbreadth; pathbreadth; path decomposition; dominating pair; AT-free graphs

Domaines: Complexité [cs.CC]; Mathématique discrète [cs.DM]

Guillaume Ducoffe, Arne Leitert. Equivalence between pathbreadth and strong pathbreadth. Discrete Applied Mathematics, 2019, 262, pp.185-188. ⟨10.1016/j.dam.2019.02.009⟩. ⟨hal-02055113⟩