In this paper we de ne the global defensive k-alliance (number) in a digraph D, and give several bounds on this parameter with characterizations of all digraphs attaining the bounds. In particular, for the case k = 1, we give a lower (an upper) bound on this parameter for directed trees (rooted trees). Moreover, the characterization of all directed trees (rooted trees) for which the equality holds is given. Finally, we show that the problem of nding the global defensive k-alliance number of a digraph is NP-hard for any suitable non-negative value of k, and in contrast with it, we also show that nding a minimum global defensive (1)-alliance for any rooted tree is polynomial-time solvable.
