Exponential speedup of fixed-parameter algorithms for classes of graphs excluding single-crossing graphs as minors

摘要

We present a fixed-parameter algorithm that constructively solves the k-dominating set problem on any class of graphs excluding a single-crossing graph ( a graph that can be drawn in the plane with at most one crossing) as a minor in O(4(9.55rootk)n(O(1))) time. Examples of such graph classes are the K-3,K-3-minor-free graphs and the K-5-minor-free graphs. 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 collection of vertex-removal problems. Our work generalizes and extends the recent results of exponential speedup in designing fixed-parameter algorithms on planar graphs due to Alber et al. to other (nonplanar) classes of graphs.