The possible cardinalities of global secure sets in cographs

Let G=(V,E) be a graph. A global secure set SD⊆V is a dominating set which also satisfies a condition that |N[X]∩SD|≥|N[X]−SD| for every subset X⊆SD. The minimum cardinality of the global secure set in the graph G is denoted by γs(G). In this paper, we introduce the notion of γs-monotone graphs. The graph G is γs-monotone if, for every k∈{γs(G),γs(G)+1,…,n}, it has a global secure set of cardinality k. We will also present the results concerning the minimum cardinality of the global secure sets in the class of cographs.