Exponential speedup of fixed-parameter algorithms on K33-minor-free or K5-minor-free graphs

Abstract

We present a fixed-parameter algorithm that constructively solves the k-dominating set problem on graphs excluding one of K5 or K 3,3 as a minor in time O(416.5√kn O(1)), which is an exponential factor faster than the previous O(2O(k)nO(1)). In fact, we present our algorithm for any H-minor-free graph where H is a single-crossing graph (can be drawn in the plane with at most one crossing) and obtain the algorithm for K3,3(K 5)-minor-free graphs as a special case. As a consequence, we extend our results to several other problems such as vertex cover, edge dominating set, independent set, clique-transversal set, kernels in digraphs, feedback vertex set and a series of vertex removal problems. Our work generalizes and extends the recent result of exponential speedup in designing fixed-parameter algorithms on planar graphs by Alber et al. to other (nonplanar) classes of graphs. © Springer-Verlag Berlin Heidelberg 2002.