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A graph $G$ is called B$_k$-VPG (resp., B$_k$-EPG), for some constant $k\geq0$, if it has a string representation on a grid such that each vertex is anorthogonal path with at most $k$ bends and two vertices are adjacent in $G$ ifand only if the corresponding strings intersect (resp., the correspondingstrings share at least one grid edge). If two adjacent strings of a B$_k$-VPGgraph intersect exactly once, then the graph is called a one-string B$_k$-VPGgraph.  In this paper, we study the Maximum Independent Set and Minimum DominatingSet problems on B$_1$-VPG and B$_1$-EPG graphs. We first give a simple $O(\logn)$-approximation algorithm for the Maximum Independent Set problem onB$_1$-VPG graphs, improving the previous $O((\log n)^2)$-approximationalgorithm of Lahiri et al. (COCOA 2015). Then, we consider the MinimumDominating Set problem. We give an $O(1)$-approximation algorithm for thisproblem on one-string B$_1$-VPG graphs, providing the first constant-factorapproximation algorithm for this problem. Moreover, we show that the MinimumDominating Set problem is APX-hard on B$_1$-EPG graphs, ruling out thepossibility of a PTAS unless P=NP. Finally, we give constant-factorapproximation algorithms for this problem on two non-trivial subclasses ofB$_1$-EPG graphs. To our knowledge, these are the first results for the MinimumDominating Set problem on B$_1$-EPG graphs, partially answering a questionposed by Epstein et al. (WADS 2013).

Approximation Algorithms for Independence and Domination on B$_1$-VPG and B$_1$-EPG Graphs