initiated the study of the mathematical properties of alliances. In particular, several bounds on the defensive alliance number were given. The particular case of global defensive alliance (generalization of dominating sets) was investigated in [@FLH3; @HHH; @SRV2] where several bounds on the global defensive alliance number were obtained. Several tight bounds on different types of alliance numbers of a graph were obtained in [@KHH; @SRV2; @S; @SGBR], namely, (global) defensive alliance number, (global) offensive alliance number and (global) dual alliance number. Moreover, alliances, as a graph-theoretic concept, have recently attracted a great deal of attention due to some interesting applications in a variety of areas, including quantitative analysis of secondary RNA structures [@HKSZ] and national defense [@P]. Besides, defensive alliances could work as a mathematical model of web communities. Adopting the definition of a Web community proposed recently in [@FLG], “a Web community is a set of web pages having more hyperlinks (in either direction) to members of the set than to non-members”. In [@CHRT], the authors use the cardinality of the strong defensive alliances by define the strong alliance polynomial of a graph. The exact defensive alliance was defined in [@C]. The alliance polynomial was introduced in [@To] and 