]{}problem is [<span style="font-variant:small-caps;">APX</span>-hard ]{}on [<span style="font-variant:small-caps;">B$_1$-EPG</span> ]{}graphs, even if only two types of paths are allowed in the input graph. Thus, there exists no PTAS for this problem on [<span style="font-variant:small-caps;">B$_1$-EPG</span> ]{}graphs unless <span style="font-variant:small-caps;">P=NP</span>.

Let us also mention that Asinowski and Ries [@AsinowskiR12] showed that the class of [<span style="font-variant:small-caps;">B$_1$-EPG</span> ]{}graphs contains a number of subclasses of chordal graphs; namely, chordal bull-free, chordal claw-free and chordal diamond-free graphs; however, we were unable to find any reference on the status of the [<span style="font-variant:small-caps;">Minimum Dominating Set</span> ]{}problem on these subclasses of chordal graphs. To show the <span style="font-variant:small-caps;">APX</span>-hardness, we give an <span style="font-variant:small-caps;">L</span>-reduction from the [<span style="font-variant:small-caps;">Minimum Vertex Cover</span> ]{}problem on graphs with maximum-degree three (which is known to be [<span style="font-variant:small-caps;">APX</span>-hard ]{} [@AlimontiK00]) to the [<span style="font-variant:small-caps;">Minimum Dominating Set</span> ]{}problem on [<span style="font-variant:small-caps;">B$_1$-EPG</span> ]{}graphs. As noted above, there is a 4-approximation algorithm for the [<span style="font-variant:small-caps;">Minimum Dominating Set</span> ]{}problem on [<span style="font-variant:small-caps;">B$_1$-EPG</span> ]{}graphs [@ButmanHLR10]. Here, we give $c$-approximation algorithms for the [<span style="font-variant:small-caps;">Minimum Dominating Set</span> ]{}problem on two non-trivial subclasses of [<span style="font-variant:small-caps;">B$_1$-EPG</span> ]{}graphs, for $c\in\{2, 3\}$. Let us defer the definition of these subclasses to Section \[subsec:MDSonBEPG\].

-   We give polynomial-time constant-factor approximation algorithms for the [<span 