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.

翻译：对于无孤立顶点的简单图 $G=(V,E)$，其协安全支配集 $D$ 满足以下两个性质：(i) $S$ 是 $G$ 的支配集；(ii) 对每个顶点 $v \in S$，存在顶点 $u \in V \setminus S$ 使得 $uv \in E$ 且 $(S \setminus \{v\}) \cup \{u\}$ 是 $G$ 的支配集。$G$ 的协安全支配集的最小基数称为 $G$ 的协安全支配数，记为 $\gamma_{cs}(G)$。最小协安全支配问题旨在找到图 $G$ 中基数为 $\gamma_{cs}(G)$ 的协安全支配集。该问题的判定版本已知对二部图、平面图和分裂图是 NP-完全的。同时，最小协安全支配问题对真区间图和余图存在高效求解算法。本文致力于研究多种重要图类以缩小最小协安全支配问题的复杂度缺口。我们证明了该问题的判定版本对圆图、双弦图、弦二部图、星凸二部图和梳凸二部图仍保持 NP-完全性。在正面结果方面，我们给出了计算链图（二部图的重要子类）协安全支配数的高效算法。此外，我们证明了该问题对有界树宽图可在线性时间内求解。进一步，我们证明了该问题的计算复杂度与支配问题存在差异。