of such an $\varepsilon$-net is not straightforward and requires a decomposition of the problem into two “one-dimensional” instances, each of which will then admit such an $\varepsilon$-net. Informally speaking, the idea is that although the [<span style="font-variant:small-caps;">Minimum Dominating Set</span> ]{}problem is not “decomposable” into horizontal and vertical instances, the $\varepsilon$-net corresponding to this problem is. Combining these nets along with the technique of Brönnimann and Goodrich [@BronnimannG95] gives us the desired result.

For [<span style="font-variant:small-caps;">B$_1$-EPG</span> ]{}graphs, the [<span style="font-variant:small-caps;">Maximum Independent Set</span> ]{}problem is known to be <span style="font-variant:small-caps;">APX</span>-hard [@BougeretBGP15], and there exists a polynomial-time 4-approximation algorithm for this problem [@EpsteinGM13]. Moreover, there exists 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], as it is known that such graphs are a subset of 2-interval graphs [@HeldtKU14]. However, we were unable to find a reference on the complexity of the [<span style="font-variant:small-caps;">Minimum Dominating Set</span> ]{}problem on [<span style="font-variant:small-caps;">B$_1$-EPG</span> ]{}graphs. We note here that every tree is a [<span style="font-variant:small-caps;">B$_1$-EPG</span> ]{}graph [@GolumbicLS09], and even permutation graphs or co-graphs are known to be [<span style="font-variant:small-caps;">B$_1$-EPG</span> ]{}[@CohenGR14], but [<span style="font-variant:small-caps;">Minimum Dominating Set</span> ]{}is polynomial-time solvable on these graphs [@graphTheoryBook].

-   We prove that the [<span style="font-variant:small-caps;">Minimum Dominating Set</span> 