  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.
