if $(u, v)\in E$.

We remark that by *intersecting* each other, we exclude the case where two paths only *touch* each other. Figure \[fig:sampleGraphs\] shows a graph with its [<span style="font-variant:small-caps;">B$_1$-VPG</span> ]{}and [<span style="font-variant:small-caps;">B$_1$-EPG</span> ]{}representations. A natural restriction on [<span style="font-variant:small-caps;">B$_1$-VPG</span> ]{}graphs can be the number of intersections allowed between every pair of paths. A string graph is called *one-string* if it has a string representation in which curves intersect at most once [@ChalopinGO10; @BiedlD15]. By combining one-string and [<span style="font-variant:small-caps;">B$_1$-VPG</span> ]{}representations, a *one-string [<span style="font-variant:small-caps;">B$_1$-VPG</span> ]{}graph* is defined as a [<span style="font-variant:small-caps;">B$_1$-VPG</span> ]{}graph in which two paths intersect each other exactly once whenever the corresponding vertices are adjacent.

![A graph on six vertices (left) with its [<span style="font-variant:small-caps;">B$_1$-VPG</span> ]{}(middle) and [<span style="font-variant:small-caps;">B$_1$-EPG</span> ]{}(right) representations. Notice that the vertices $e$ and $f$ are not adjacent in the [<span style="font-variant:small-caps;">B$_1$-EPG</span> ]{}representation as they only share a grid node (but not a grid edge).[]{data-label="fig:sampleGraphs"}](finalSampleGraphs){width="90.00000%"}

In this paper, we study two classical graph problems; namely, the [<span style="font-variant:small-caps;">Maximum Independent Set</span> ]{}and [<span style="font-variant:small-caps;">Minimum Dominating Set</span> ]{}problems, on [<span style="font-variant:small-caps;">B$_1$-EPG</span> ]{}and [<span style="font-variant:small-caps;">B$_1$-VPG</span> ]{}graphs. Let $G=(V, E)$ be an unweighted, undirected graph. A set $S\subseteq V$ is an independent set if no two vertices 