On the complement graph and defensive kk-alliances

In this paper, we obtain several tight bounds of the defensive kk-alliance number in the complement graph from other parameters of the graph. In particular, we investigate the relationship between the alliance numbers of the complement graph and the minimum and maximum degree, the domination number and the isoperimetric number of the graph. Moreover, we prove the NP-completeness of the decision problem underlying the defensive kk-alliance number.