Given a graph G with vertex set V, a set \(D\subseteq V\) is a dominating set of G if for each vertex \(u\in V\setminus D\) , there is a vertex \(v\in D\) adjacent to u. A secure dominating set of G is a dominating set S of G with the property that for each vertex \(u\in V\setminus S\) , there is a vertex \(v\in S\) adjacent to u such that \((S\cup \{u\})\setminus \{v\}\) is a dominating set of G. The minimum dominating set (or, for short, MDS) (resp. minimum secure dominating set (or, for short, MSDS)) problem is to find an MDS (resp. MSDS) in a given graph. In this paper, we first discuss the NP-completeness and APX-hardness of the MSDS problem on EPG graphs and VPG graphs. Then we provide PTASes for the MDS problem and the MSDS problem on EPG graphs and VPG graphs, where the length of the horizontal part and vertical part of each path is at most c, respectively, where \(c\ge 1\) is a constant. Thus the answer is negative for the question that Bessy et al. (2020) asked if the MDS problem is APX-complete on \(\{\lrcorner \}\) -EPG graphs, where the horizontal and the vertical part of each path are bounded length.
