Bounds on Directed star arboricity in some digraph classes

A galaxy is a forest of directed stars. The notion of galaxy can be related to Facility Location problems as well as wavelength assignment problems in optical networks. Amini et al. [Combinatorics, Probability & Computing, 19(2):161-182, 2010.] and Gonçalves et al. [Discrete Applied Mathematics, 160(6):744-754, 2012.] gave bounds on the minimum number of galaxies needed to cover the arcs of a digraph D, called directed star arboricity (dst(D)). They conjectured that those bounds could be improved such that dst(D)≤Δ(D), for Δ(D)≥3 and dst(D)≤2Δ+(D) for Δ+(D)≥2. In this work, we study the directed star arboricity in two non-trivial digraph classes: k-degenerate digraphs and tournaments.