  A subset $D \subseteq V $of a graph $G = (V, E)$ is a $(1, j)$-set if every
vertex $v \in V \setminus D$ is adjacent to at least $1$ but not more than $j$
vertices in D. The cardinality of a minimum $(1, j)$-set of $G$, denoted as
$\gamma_{(1,j)} (G)$, is called the $(1, j)$-domination number of $G$. Given a
graph $G = (V, E)$ and an integer $k$, the decision version of the $(1, j)$-set
problem is to decide whether $G$ has a $(1, j)$-set of cardinality at most $k$.
In this paper, we first obtain an upper bound on $\gamma_{(1,j)} (G)$ using
probabilistic methods, for bounded minimum and maximum degree graphs. Our bound
is constructive, by the randomized algorithm of Moser and Tardos [MT10], We
also show that the $(1, j)$- set problem is NP-complete for chordal graphs.
Finally, we design two algorithms for finding $\gamma_{(1,j)} (G)$ of a tree
and a split graph, for any fixed $j$, which answers an open question posed in
[CHHM13].
