Spanning galaxies in digraphs

A star is an arborescence in which the root dominates all the other vertices. A galaxy is a vertex-disjoint union of stars. The directed star arboricity of a digraph D, denoted by dst ( D ) , is the minimum number of galaxies needed to cover A ( D ) . In this paper, we show that dst ( D ) ⩽ Δ ( D ) + 1 and that if D is acyclic then dst ( D ) ⩽ Δ ( D ) . These results are proved by considering the existence of spanning galaxies in digraphs. Thus, we study the problem of deciding whether a digraph D has a spanning galaxy or not. We show that it is NP-complete (even when restricted to acyclic digraphs) but that it becomes polynomial-time solvable when restricted to strongly connected digraphs.