Monotonicity and expansion of global secure sets

In this paper we investigate two problems, namely finding a global secure set of fixed cardinality and its expansion. For a given graph G = ( V , E ) , a global secure set S D ⊆ V is a dominating set such that for every subset X ⊆ S D , | N [ X ] ∩ S D | ≥ | N [ X ] − S D | . Moreover, we say that a global secure set S D ⊂ V is expandable if there exists a vertex v ∈ ( V − S D ) such that a set S D ′ = S D ∪ { v } is a global secure set.