The possible cardinalities of global secure sets in cographs

Let G = ( V , E ) be a graph. A global secure set S D ⊆ V is a dominating set which also satisfies a condition that | N [ X ] ∩ S D | ≥ | N [ X ] − S D | for every subset X ⊆ S D . The minimum cardinality of the global secure set in the graph G is denoted by γ s ( G ) . In this paper, we introduce the notion of γ s -monotone graphs. The graph G is γ s -monotone if, for every k ∈ { γ s ( G ) , γ s ( 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.