Intractability of Clique-Width Parameterizations

We show that Edge Dominating Set, Hamiltonian Cycle, and Graph Coloring are W [1]-hard parameterized by clique-width. It was an open problem, explicitly mentioned in several papers, whether any of these problems is fixed parameter tractable when parameterized by the clique-width, that is, solvable in time g(k) · nO(1) on n-vertex graphs of clique-width k, where g is some function of k only. Our results imply that the running time O(nf(k)) of many clique-width based algorithms is essentially the best we can hope for (up to a widely believed assumption from parameterized complexity, namely FPT 6= W [1]).