A set $D\subseteq V$ of a graph $G=(V,E)$ is a dominating set of G if every vertex $v\in V\setminus D$ is adjacent to at least one vertex in D. A set $S\subseteq V$ is a co-secure dominating set (CSDS) of a graph G if S is a dominating set of G and for each vertex $u\in S$ there exists a vertex $v\in V\setminus S$ such that $uv\in E$ and $(S\setminus \{u\})\cup \{v\}$ is a dominating set of G. The minimum cardinality of a co-secure dominating set of G is the co-secure domination number and it is denoted by ${\gamma }_{cs}(G)$. Given a graph $G=(V,E)$, the minimum co-secure dominating set problem (Min Co-secure Dom) is to find a co-secure dominating set of minimum cardinality. In this paper, we strengthen the inapproximability result of Min Co-secure Dom for general graphs by showing that this problem can not be approximated within a factor of $(1-\epsilon )\ln |V|$ for perfect elimination bipartite graphs and star convex bipartite graphs unless P=NP. On the positive side, we show that Min Co-secure Dom can be approximated within a factor of $O(\ln |V|)$ for any graph G with $\delta (G)\ge 2$. For 3-regular and 4-regular graphs, we show that Min Co-secure Dom is approximable within a factor of $\frac{8}{3}$ and $\frac{10}{3}$, respectively. Furthermore, we prove that Min Co-secure Dom is APX-complete for 3-regular graphs.

一套$D\subseteq V$图表的$G=（V,乙）$是G的支配集，如果每个顶点$v\epsilon V\setminus D$与D中的至少一个顶点相邻。一套$S\subseteq V$是图G的共安全支配集 ( CSDS )，如果S是G的支配集并且对于每个顶点$你\epsilon S$存在一个顶点$v\epsilon V\setminus S$这样$你v\epsilon 乙$和$（S\setminus \{你\}）\cup \{v\}$是G的支配集。G的共安全支配集的最小基数是共安全支配数，它表示为${\gamma }_{Cs}（G）$。给定一个图$G=（V,乙）$，最小共安全支配集问题（Min Co-secure Dom）就是找到一个最小基数的共安全支配集。在本文中，我们通过证明这个问题不能在一个因子内近似来加强一般图的最小共同安全Dom的不可近似性结果$（1-\epsilon ）\mathrm{因}|V|$对于完美消除二部图和星凸二部图，除非P=NP。从积极的一面来看，我们表明Min Co-secure Dom可以在一个因子内近似$氧（\mathrm{因}|V|）$对于任意图G$\delta （G）\ge 2$。对于 3-正则图和 4-正则图，我们表明Min Co-secure Dom近似于$\frac{8}{3}$和$\frac{10}{3}$， 分别。此外，我们证明Min Co-secure Dom对于 3-正则图是APX完备的。