Hitting forbidden subgraphs in graphs of bounded treewidth

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.