On the other hand, if the input graph is planar, the problem remains NP-hard, but a fixed-parameter algorithm is known. In fact, the problem has a kernel of size linear in "k", and running times that are exponential in and cubic in "n" may be obtained by applying dynamic programming to a branch-decomposition of the kernel. More generally, the dominating set problem and many variants of the problem are fixed-parameter tractable when parameterized by both the size of the dominating set and the size of the smallest forbidden complete bipartite subgraph; that is, the problem is FPT on biclique-free graphs, a very general class of sparse graphs that includes the planar graphs.