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abstract: |
    A graph $G$ is called *B$_k$-VPG* (resp., *B$_k$-EPG*), for some constant $k\geq 0$, if it has a string representation on a grid such that each vertex is an orthogonal path with at most $k$ bends and two vertices are adjacent in $G$ if and only if the corresponding strings intersect (resp., the corresponding strings share at least one grid edge). If two adjacent strings of a B$_k$-VPG graph intersect exactly once, then the graph is called a *one-string* B$_k$-VPG graph.

    In this paper, we study 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$_1$-VPG</span> ]{}and [<span style="font-variant:small-caps;">B$_1$-EPG</span> ]{}graphs. We first give a simple $O(\log n)$-approximation algorithm for the [<span style="font-variant:small-caps;">Maximum Independent Set</span> ]{}problem on [<span style="font-variant:small-caps;">B$_1$-VPG</span> ]{}graphs, improving the previous $O((\log n)^2)$-approximation algorithm of Lahiri et al. [@LahiriMS15]. Then, we consider the [<span style="font-variant:small-caps;">Minimum Dominating Set</span> ]{}problem. We give an $O(1)$-approximation algorithm for this problem on one-string [<span style="font-variant:small-caps;">B$_1$-VPG</span> ]{}graphs, providing the first constant-factor approximation algorithm for this problem. Moreover, we show that the [<span style="font-variant:small-caps;">Minimum Dominating Set</span> ]{}problem is [<span style="font-variant:small-caps;">APX</span>-hard ]{}on [<span style="font-variant:small-caps;">B$_1$-EPG</span> ]{}graphs, ruling out the possibility of a PTAS unless <span style="font-variant:small-caps;">P=NP</span>. Finally, we give constant-factor approximation algorithms 