Some new results on the security number in the Cartesian product of graphs

A set of vertices S ⊆ V (G) is secure if it is possible to thwart an assault on any vertex set X ⊆ S. That is, a set S is secure if and only if it can fight off every attack. The security number of G, s(G), is the minimum cardinality of a secure set in G. In this paper, we initiated a study of security in the Cartesian product of graphs and obtained a characterization for s(G□H) = 2 and 3. We have also improved the existing upper bound for s(G□H) for some classes of graphs.