Connected minimum secure-dominating sets in grids

For any (finite simple) graph G the secure domination number of G satisfies γs(G)≥|V(G)|2. Here we find a secure-dominating set S in G such that |S|=⌈|V(G)|2⌉ in all cases when G is a grid, and in the majority of cases when G is a cylindrical or toroidal grid. In all such cases, S satisfies the additional requirement that G[S] is connected. We make note that the concept of secure-dominating sets considered in this paper is quite different from the other secure domination currently of interest.1