Domination and fractional domination in digraphs

In this paper, we investigate the relation between the (fractional) domination number of a digraph $G$ and the independence number of its underlying graph, denoted by $\alpha(G)$. More precisely, we prove that every digraph $G$ has fractional domination number at most $2\alpha(G)$, and every directed triangle-free digraph $G$ has domination number at most $\alpha(G)\cdot \alpha(G)!$. The first bound is sharp.