  We study the complexity of a generic hitting problem H-Subgraph Hitting,
where given a fixed pattern graph $H$ and an input graph $G$, the task is to
find a set $X \subseteq V(G)$ of minimum size that hits all subgraphs of $G$
isomorphic to $H$. In the colorful variant of the problem, each vertex of $G$
is precolored with some color from $V(H)$ and we require to hit only
$H$-subgraphs with matching colors. Standard techniques shows that for every
fixed $H$, the problem is fixed-parameter tractable parameterized by the
treewidth of $G$; however, it is not clear how exactly the running time should
depend on treewidth. For the colorful variant, we demonstrate matching upper
and lower bounds showing that the dependence of the running time on treewidth
of $G$ is tightly governed by $\mu(H)$, the maximum size of a minimal vertex
separator in $H$. That is, we show for every fixed $H$ that, on a graph of
treewidth $t$, the colorful problem can be solved in time
$2^{\mathcal{O}(t^{\mu(H)})}\cdot|V(G)|$, but cannot be solved in time
$2^{o(t^{\mu(H)})}\cdot |V(G)|^{O(1)}$, assuming the Exponential Time
Hypothesis (ETH). Furthermore, we give some preliminary results showing that,
in the absence of colors, the parameterized complexity landscape of H-Subgraph
Hitting is much richer.
