  For a simple graph $G=(V,E)$ without any isolated vertex, a cosecure
dominating set $D$ of $G$ satisfies the following two properties (i) $S$ is a
dominating set of $G$, (ii) for every vertex $v \in S$ there exists a vertex $u
\in V \setminus S$ such that $uv \in E$ and $(S \setminus \{v\}) \cup \{u\}$ is
a dominating set of $G$. The minimum cardinality of a cosecure dominating set
of $G$ is called cosecure domination number of $G$ and is denoted by
$\gamma_{cs}(G)$. The Minimum Cosecure Domination problem is to find a cosecure
dominating set of a graph $G$ of cardinality $\gamma_{cs}(G)$. The decision
version of the problem is known to be NP-complete for bipartite, planar, and
split graphs. Also, it is known that the Minimum Cosecure Domination problem is
efficiently solvable for proper interval graphs and cographs.
  In this paper, we work on various important graph classes in an effort to
reduce the complexity gap of the Minimum Cosecure Domination problem. We show
that the decision version of the problem remains NP-complete for circle graphs,
doubly chordal graphs, chordal bipartite graphs, star-convex bipartite graphs
and comb-convex bipartite graphs. On the positive side, we give an efficient
algorithm to compute the cosecure domination number of chain graphs, which is
an important subclass of bipartite graphs. In addition, we show that the
problem is linear-time solvable for bounded tree-width graphs. Further, we
prove that the computational complexity of this problem varies from the
domination problem.
