On $k$-vertex-edge domination of graph

Let $G=(V,E)$ be a simple undirected graph. The open neighbourhood of a vertex $v$ in $G$ is defined as $NG(v)={u\in V~|~ uv\in E}$; whereas the closed neighbourhood is defined as $NG[v]= NG(v)\cup {v}$. For an integer $k$, a subset $D\subseteq V$ is called a $k$-vertex-edge dominating set of $G$ if for every edge $uv\in E$, $|(NG[u]\cup N_G[v]) \cap D|\geq k$. In $k$-vertex-edge domination problem, our goal is to find a $k$-vertex-edge dominating set of minimum cardinality of an input graph $G$. In this paper, we first prove that the decision version of $k$-vertex-edge domination problem is NP-complete for chordal graphs. On the positive side, we design a linear time algorithm for finding a minimum $k$-vertex-edge dominating set of tree. We also prove that there is a $O(\log(\Delta(G)))$-approximation algorithm for this problem in general graph $G$, where $\Delta(G)$ is the maximum degree of $G$. Then we show that for a graph $G$ with $n$ vertices, this problem cannot be approximated within a factor of $(1-\epsilon) \ln n$ for any $\epsilon >0$ unless $NP\subseteq DTIME(|V|^{{O(\log\log|V|)})$.} Finally, we prove that it is APX-complete for graphs with bounded degree $k+3$.