Exponential Speedup of Fixed-Parameter Algorithms for Classes of Graphs Excluding Single-Crossing Graphs as Minors

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 $O(4^{9.55\sqrt{k}}n^{O(1)})$ time. Examples of such graph classes are the $K_{3,3}$-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.