Global secure sets of grid-like graphs

Let G = (V. E) be a graph and S subset of V. The set S is a secure set if for all X subset of S, vertical bar N vertical bar X vertical bar boolean AND S vertical bar >= vertical bar N vertical bar X vertical bar-S vertical bar, and S is a global secure set if S is a secure set and a dominating set. The cardinality of a minimum global secure set of G is the global security number of G, denoted gamma(s)(G). The sets studied in this paper are different from secure dominating sets studied in Cockayne et al. (2003) 131, Grobler and Mynhardt (2009) [81, or Klostermeyer and Mynhardt (2008) [13], which are also denoted by gamma(s). In this paper, we provide results on the global security numbers of paths, cycles and their Cartesian products. Published by Elsevier B.V.