Given a graph G with vertex set V, a set $S\subseteq V$ is a secure dominating set of G if S is a dominating set of G and if for every vertex $u\in V\setminus S$, there exists a vertex $v\in S$ adjacent to u such that $(S\cup \{u\})\setminus \{v\}$ is a dominating set of G.The minimum secure dominating set (or, for short, MSDS) problem asks to find an MSDS in a given graph. In this paper, we first show that the decision version of the MSDS problem is NP-complete in unit disk graphs, even in grid graphs. Secondly, we give an $O(n+m)$ time t-approximation algorithm for the MSDS problem in several geometric intersection graphs which are $K_{1,t}$-free for some integer $t\ge 3$. Finally, we propose a PTAS for the MSDS problem in unit disk graphs.

给定一个带有顶点集V的图G，一个集合$S\subseteq V$是G的安全支配集，如果S是G的支配集并且对于每个顶点$你\epsilon V\setminus S$，存在一个顶点$v\epsilon S$与你相邻，使得$（S\cup \{你\}）\setminus \{v\}$是G的支配集。最小安全支配集（或简称 MSDS）问题要求在给定图中找到 MSDS。在本文中，我们首先证明 MSDS 问题的决策版本在单位圆盘图中（甚至在网格图中）是 NP 完全的。其次，我们给出一个$氧（n+米）$几个几何相交图中 MSDS 问题的时间t近似算法$K_{1,t}$-对于某些整数来说是免费的$t\ge 3$。最后，我们针对单位圆盘图中的 MSDS 问题提出了 PTAS。