in $S$ are adjacent, while $S$ is a dominating set if every vertex in $V\setminus S$ is adjacent to some vertex in $S$. The objective of the [<span style="font-variant:small-caps;">Maximum Independent Set</span> ]{}problem is to compute a maximum-cardinality independent set of $G$, whereas the goal in [<span style="font-variant:small-caps;">Minimum Dominating Set</span> ]{}is to minimize the size of the dominating set $S$. [<span style="font-variant:small-caps;">Maximum Independent Set</span> ]{}and [<span style="font-variant:small-caps;">Minimum Dominating Set</span> ]{}are two fundamental optimization problems in graph theory, which arise in many applications such as wireless sensor networks, scheduling and resource allocation; both problems are well known to be <span style="font-variant:small-caps;">NP</span>-hard.

It is known that every graph has an EPG representation [@GolumbicLS09]. Heldt et al. [@HeldtKU14] showed that every graph of maximum degree $\Delta$ has a B$_\Delta$-EPG representation. Moreover, Heldt et al. [@HeldtKU14a] proved that the class of B$_4$-EPG graphs contains all planar graphs; it is however still open whether $k=4$ is best possible. They also proved a conjecture of Biedl and Stern [@BiedlS10] affirmatively stating that all outerplanar graphs are B$_2$-EPG graphs. For the case of [<span style="font-variant:small-caps;">B$_1$-EPG</span> ]{}graphs, Golumbic et al. [@GolumbicLS09] showed that every tree is a [<span style="font-variant:small-caps;">B$_1$-EPG</span> ]{}graph, and a characterization of the subfamily of co-graphs that 