$(1,j)$-set problem in graphs

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].