The algorithm and complexity of secure domination in 3-dimensional box graphs

Given a graph $G$ with vertex set $V$, a secure dominating set of $G$ is a set $S\subseteq V$ with the property that for each vertex $u\in V\setminus S$, there is a vertex $v\in S$ adjacent to $u$ such that $(S\cup \left\{u\right\})\setminus \left\{v\right\}$ 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, firstly, we show that the MSDS problem is APX-hard in $d$-box graphs for any fixed integer $d\ge 2$. Secondly, we obtain a PTAS for the MSDS problem which is $(1+\frac{21}{k})$-approximation (resp. $(1+\frac{6}{k})$-approximation) and runs in $n^{O\left(k^3\right)}$ (resp. $n^{O\left(k^2\right)}$) time for unit cube graphs and unit ball graphs (resp. unit square graphs and unit disk graphs). Thirdly, we give a dynamic programming algorithm for the MSDS problem in unit-height box graphs (resp. unit-height rectangle graphs), where the centers of all boxes (resp. rectangles) are inside a column of length $l$ and width $w$ for some integers $l,w\ge 1$ (resp. a strip of width $w$ for some integer $w\ge 1$). Finally, with the help of the dynamic programming algorithm, we improve the PTAS to $(1+\frac{6}{k})$-approximation (resp. $(1+\frac{1}{k})$-approximation) and $n^{O\left(k^2\right)}$ (resp. $n^{O\left(k\right)}$) time for unit cube graphs and unit ball graphs (resp. unit square graphs and unit disk graphs).