  A subset $S \subseteq V$ in a graph $G = (V,E)$ is called a $[1, k]$-set, if
for every vertex $v \in V \setminus S$, $1 \leq | N_G(v) \cap S | \leq k$. The
$[1,k]$-domination number of $G$, denoted by $\gamma_{[1, k]}(G)$ is the size
of the smallest $[1,k]$-sets of $G$. A set $S'\subseteq V(G)$ is called a total
$[1,k]$-set, if for every vertex $v \in V$, $1 \leq | N_G(v) \cap S | \leq k$.
If a graph $G$ has at least one total $[1, k]$-set then the cardinality of the
smallest such set is denoted by $\gamma_{t[1, k]}(G)$. We consider $[1,
k]$-sets that are also independent. Note that not every graph has an
independent $[1, k]$-set. For graphs having an independent $[1, k]$-set, we
define $[1, k]$-independence numbers which is denoted by $\gamma_{i[1, k]}(G)$.
In this paper, we investigate the existence of $[1,k]$-sets in lexicographic
products $G\circ H$. Furthermore, we completely characterize graphs which their
lexicographic product has at least one total $[1,k]$-set. Also, we determine
$\gamma_{[1, k]}(G\circ H)$, $\gamma_{t[1, k]}(G\circ H)$ and $\gamma_{i[1,
k]}(G\circ H)$. Finally, we show that finding smallest total $[1, k]$-set is
$NP$-complete.
