k-secure sets and k-security number of a graph

Let G=(V, E) be a simple connected graph. A nonempty set S\subseteq V is a secure set if every attack on S is defendable. In this paper, k-secure sets are introduced as a generalization of secure sets. For any integer k\geq 0, a nonempty subset S of V is a $k-secure set if, for each attack on S, there is a defense of $S$ such that for every $v\in S$, the defending set of $v$ contains at least $k$ more elements than that of the attacking set of $v$, whenever the vertex $v$ has neighbors outside $S$. The cardinality of a minimum $k$-secure set in $G$ is the $k$-security number of $G$. Some properties of $k$-secure sets are discussed and a characterization of $k$-secure sets is obtained. Also, 1-security numbers of certain classes of graphs are determined.