Let $G=(V,E)$ be a simple undirected graph. The open neighbourhood of a vertex $v$ in $G$ is defined as $N_G(v)=\{u\in V~|~ uv\in E\}$; whereas the closed neighbourhood is defined as $N_G[v]= N_G(v)\cup \{v\}$. For an integer $k$, a subset $D\subseteq V$ is called a $k$-vertex-edge dominating set of $G$ if for every edge $uv\in E$, $|(N_G[u]\cup N_G[v]) \cap D|\geq k$. 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 $k+3$.

翻译：设$G=(V,E)$为一个简单无向图。顶点$v$在$G$中的开邻域定义为$N_G(v)=\{u\in V~|~ uv\in E\}$，而闭邻域定义为$N_G[v]= N_G(v)\cup \{v\}$。对于整数$k$，子集$D\subseteq V$称为$G$的一个$k$顶点-边支配集，当且仅当对每条边$uv\in E$，有$|(N_G[u]\cup N_G[v]) \cap D|\geq k$。在$k$顶点-边支配问题中，我们的目标是寻找输入图$G$的最小基数$k$顶点-边支配集。本文首先证明了$k$顶点-边支配问题的判定版本对弦图是NP完全的。在积极方面，我们设计了一个线性时间算法，用于寻找树的最小$k$顶点-边支配集。同时我们证明，对于一般图$G$，该问题存在一个$O(\log(\Delta(G)))$-近似算法，其中$\Delta(G)$是$G$的最大度。接着我们证明，对于具有$n$个顶点的图$G$，除非$NP\subseteq DTIME(|V|^{O(\log\log|V|)})$，否则该问题无法在$(1-\epsilon) \ln n$因子内近似（对任意$\epsilon >0$）。最后，我们证明该问题在最大度为$k+3$的有界度图上是APX完全的。