Global defensive alliances of trees and Cartesian product of paths and cycles

Given a graph G, a defensive alliance of G is a set of vertices S⊆V(G) satisfying the condition that for each v∈S, at least half of the vertices in the closed neighborhood of v are in S. A defensive alliance S is called global if every vertex in V(G)−S is adjacent to at least one member of the defensive alliance S. The global defensive alliance number of G, denoted γa(G), is the minimum size around all the global defensive alliances of G. In this paper, we present an efficient algorithm to determine the global defensive alliance numbers of trees, and further give formulas to decide the global defensive alliance numbers of complete k-ary trees for k=2,3,4. We also establish upper bounds and lower bounds for γa(Pm×Pn),γa(Cm×Pn) and γa(Cm×Cn), and show that the bounds are sharp for certain m,n.