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

Let g=(v, e) be a simple connected graph. a nonempty set ssubseteq 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 kgeq 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 $vin s$, 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.