  We consider the Minimum Dominating Set (MDS) problem on the intersection
graphs of geometric objects. Even for simple and widely-used geometric objects
such as rectangles, no sub-logarithmic approximation is known for the problem
and (perhaps surprisingly) the problem is NP-hard even when all the rectangles
are "anchored" at a diagonal line with slope -1 (Pandit, CCCG 2017). In this
paper, we first show that for any $\epsilon>0$, there exists a
$(2+\epsilon)$-approximation algorithm for the MDS problem on
"diagonal-anchored" rectangles, providing the first $O(1)$-approximation for
the problem on a non-trivial subclass of rectangles. It is not hard to see that
the MDS problem on "diagonal-anchored" rectangles is the same as the MDS
problem on "diagonal-anchored" L-frames: the union of a vertical and a
horizontal line segment that share an endpoint. As such, we also obtain a
$(2+\epsilon)$-approximation for the problem with "diagonal-anchored" L-frames.
On the other hand, we show that the problem is APX-hard in case the input
L-frames intersect the diagonal, or the horizontal segments of the L-frames
intersect a vertical line. However, as we show, the problem is linear-time
solvable in case the L-frames intersect a vertical as well as a horizontal
line. Finally, we consider the MDS problem in the so-called "edge intersection
model" and obtain a number of results, answering two questions posed by Mehrabi
(WAOA 2017).
