A dominating set $S$ is an Isolate Dominating Set (IDS) if the induced subgraph $G[S]$ has at least one isolated vertex. In this paper, we initiate the study of new domination parameter called, isolate secure domination. An isolate dominating set $S\subseteq V$ is an isolate secure dominating set (ISDS), if for each vertex $u \in V \setminus S$, there exists a neighboring vertex $v$ of $u$ in $S$ such that $(S \setminus \{v\}) \cup \{u\}$ is an IDS of $G$. The minimum cardinality of an ISDS of $G$ is called as an isolate secure domination number, and is denoted by $\gamma_{0s}(G)$. Given a graph $ G=(V,E)$ and a positive integer $ k,$ the ISDM problem is to check whether $ G $ has an isolate secure dominating set of size at most $ k.$ We prove that ISDM is NP-complete even when restricted to bipartite graphs and split graphs. We also show that ISDM can be solved in linear time for graphs of bounded tree-width.

中文翻译：

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图中隔离安全支配的算法复杂性

如果诱导子图 $G[S]$ 至少有一个孤立的顶点，则支配集 $S$ 是一个孤立支配集（IDS）。在本文中，我们开始研究称为隔离安全控制的新控制参数。一个隔离支配集 $S\subseteq V$ 是一个隔离安全支配集（ISDS），如果对于每个顶点 $u \in V \setminus S$，在 $S$ 中都存在一个 $u$ 的相邻顶点 $v$使得 $(S \setminus \{v\}) \cup \{u\}$ 是 $G$ 的 IDS。$G$ 的 ISDS 的最小基数称为隔离安全支配数，用 $\gamma_{0s}(G)$ 表示。给定一个图 $ G=(V,E)$ 和一个正整数 $ k,$ ISDM 问题是检查 $ G $ 是否有一个最大大小为 $ k 的隔离安全支配集。 我们证明 ISDM 是 NP - 即使仅限于二部图和分割图，也能完成。