Global secure sets of grid-like graphs

Let G=(V,E)G=(V,E) be a graph and S⊆VS⊆V. The set SS is a secure set   if ∀X⊆S,|N[X]∩S|≥|N[X]−S|∀X⊆S,|N[X]∩S|≥|N[X]−S|, and SS is a global secure set if SS is a secure set and a dominating set. The cardinality of a minimum global secure set of GG is the global security number of GG, denoted γs(G)γs(G). The sets studied in this paper are different from secure dominating sets studied in Cockayne et al. (2003)  [3], Grobler and Mynhardt (2009) [8], or Klostermeyer and Mynhardt (2008) [13], which are also denoted by γsγs.In this paper, we provide results on the global security numbers of paths, cycles and their Cartesian products.