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 $\delta(G)\geq 2$. For $3$-regular and $4$-regular graphs, we show that Min Co-secure Dom is approximable within a factor of $\dfrac{8}{3}$ and $\dfrac{10}{3}$, respectively. Furthermore, we prove that Min Co-secure Dom is APX-complete for $3$-regular graphs.

翻译：设图$G=(V, E)$的子集$D \subseteq V$是$G$的一个支配集，若每个顶点$v\in V\setminus D$至少与$D$中的一个顶点相邻。子集$S \subseteq V$是图$G$的一个共安全支配集（CSDS），若$S$是$G$的一个支配集，且对于每个顶点$u \in S$，存在一个顶点$v \in V\setminus S$，使得$uv \in E$且$(S\setminus \{u\}) \cup \{v\}$是$G$的一个支配集。$G$的最小共安全支配集的基数称为共安全支配数，记作$\gamma_{cs}(G)$。给定图$G=(V, E)$，最小共安全支配集问题（Min Co-secure Dom）是寻找一个基数最小的共安全支配集。本文中，我们强化了一般图的Min Co-secure Dom问题的不可近似性结果，表明对于完美消除二部图和星凸二部图，除非P=NP，否则该问题无法在因子$(1- \epsilon)\ln |V|$内近似。在积极方面，我们证明对于任意满足$\delta(G)\geq 2$的图$G$，Min Co-secure Dom可以在因子$O(\ln |V|)$内近似。对于$3$-正则图和$4$-正则图，我们证明Min Co-secure Dom分别可在因子$\dfrac{8}{3}$和$\dfrac{10}{3}$内近似。此外，我们证明对于$3$-正则图，Min Co-secure Dom是APX完全的。