The security number of strong grid-like graphs

The concepts of secure sets and security number in graphs were introduced first by Brigham et al. in 2007 as a generalization of the concept of alliances in graphs. Defensive alliances are related to the defense of a single vertex. But, in a general realistic settings, alliances should be formed so that any attack on the entire alliance or any subset of the alliance can be defended. In this sense, secure sets represent an attempt to develop a model of this situation. Given a graph G = (V,E) and a set of vertices S ⊆ V of G, the set S is a secure set if it can defend every attack from vertices outside of S, according to an appropriate definition of “attack” and “defense”. The minimum cardinality of a secure set in G is the security number s(G). In this article we obtain the security number of grid-like graphs, obtained as the strong product graph of paths and cycles (grids, cylinders and torus). Specifically we obtain that for any two positive integers m,n ≥ 4, s(Pm Pn) = min{m,n, 8}, s(Pm Cn) = min{2m,n, 16} and s(Cm Cn) = min{2m, 2n, 32}.