We describe an algorithm for the dominating set problem with time complexity O((4g+41)^kn^2) for graphs of bounded genus g, where k is the size of the set. It has previously been shown that this problem is fixed parameter tractable for planar graphs. Our method is a refinement of the earlier techniques.
The following paper was presented at SWAT 2002 and contains the main result for any graph of bounded genus.
This paper also contains a simpler proof for the previously known O(8^kn^2) result for planar graphs.