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)G=(V,E), a global secure set SD⊆VSD⊆V is a dominating set such that for every subset X⊆SDX⊆SD, |N[X]∩SD|≥|N[X]−SD||N[X]∩SD|≥|N[X]−SD|. Moreover, we say that a global secure set SD⊂VSD⊂V is expandable if there exists a vertex v∈(V−SD)v∈(V−SD) such that a set SD′=SD∪{v}SD′=SD∪{v} is a global secure set.