  We consider the problem of finding a lowest cost dominating set in a given
disk graph containing $n$ disks. The problem has been extensively studied on
subclasses of disk graphs, yet the best known approximation for disk graphs has
remained $O(\log n)$ -- a bound that is asymptotically no better than the
general case. We improve the status quo in two ways: for the unweighted case,
we show how to obtain a PTAS using the framework recently proposed
(independently)by Mustafa and Ray [SoCG 09] and by Chan and Har-Peled [SoCG
09]; for the weighted case where each input disk has an associated rational
weight with the objective of finding a minimum cost dominating set, we give a
randomized algorithm that obtains a dominating set whose weight is within a
factor $2^{O(\log^* n)}$ of a minimum cost solution, with high probability --
the technique follows the framework proposed recently by Varadarajan [STOC 10].
