  We study the Directed Feedback Vertex Set problem parameterized by the
treewidth of the input graph. We prove that unless the Exponential Time
Hypothesis fails, the problem cannot be solved in time $2^{o(t\log t)}\cdot
n^{\mathcal{O}(1)}$ on general directed graphs, where $t$ is the treewidth of
the underlying undirected graph. This is matched by a dynamic programming
algorithm with running time $2^{\mathcal{O}(t\log t)}\cdot n^{\mathcal{O}(1)}$.
On the other hand, we show that if the input digraph is planar, then the
running time can be improved to $2^{\mathcal{O}(t)}\cdot n^{\mathcal{O}(1)}$.
