On spanning galaxies in digraphs

In a directed graph, a star is an arborescence with at least one arc, in which the root dominates all the other vertices. A galaxy is a vertex-disjoint union of stars. In this paper, we consider the Spanning Galaxy problem of deciding whether a digraph D has a spanning galaxy or not. We show that although this problem is NP-complete (even when restricted to acyclic digraphs), it becomes polynomial-time solvable when restricted to strong digraphs. In fact, we prove that restricted to this class, the Spanning Galaxy problem it is equivalent to the problem of deciding if a strong digraph has a strong digraph with an even number of vertices. We then show a polynomial time algorithm to solve this problem. We also consider some parameterized version of the Spanning Galaxy problem. Finally, we improve some results concerning the notion of directed star arboricity of a digraph D, which is the minimum number of galaxies needed to cover all the arcs of D. We show in particular that dst(D) ≤ ∆(D) + 1 for every digraph D and that dst(D) ≤ ∆(D) for every acyclic digraph D.