It has long been known that Feedback Vertex Set can be solved in time $2^{mathcal{O}(wlog w)}n^{mathcal{O}(1)}$ on $n$-vertex graphs of treewidth $w$, but it was only recently that this running time was improved to $2^{mathcal{O}(w)}n^{mathcal{O}(1)}$, that is, to single-exponential parameterized by treewidth. We investigate which generalizations of Feedback Vertex Set can be solved in a similar running time. Formally, for a class $mathcal{P}$ of graphs, the Bounded $mathcal{P}$-Block Vertex Deletion problem asks, given a graph~$G$ on $n$ vertices and positive integers~$k$ and~$d$, whether $G$ contains a set~$S$ of at most $k$ vertices such that each block of $G-S$ has at most $d$ vertices and is in $mathcal{P}$. Assuming that $mathcal{P}$ is recognizable in polynomial time and satisfies a certain natural hereditary condition, we give a sharp characterization of when single-exponential parameterized algorithms are possible for fixed values of $d$: if $mathcal{P}$ consists only of chordal graphs, then the problem can be solved in time $2^{mathcal{O}(wd^2)} n^{mathcal{O}(1)}$, and if $mathcal{P}$ contains a graph with an induced cycle of length $ellge 4$, then the problem is not solvable in time $2^{o(wlog w)} n^{mathcal{O}(1)}$ even for fixed $d=ell$, unless the ETH fails. We also study a similar problem, called Bounded $mathcal{P}$-Component Vertex Deletion, where the target graphs have connected components of small size rather than blocks of small size, and we present analogous results. For this problem, we also show that if $d$ is part of the input and $mathcal{P}$ contains all chordal graphs, then it cannot be solved in time $f(w)n^{o(w)}$ for some function $f$, unless the ETH fails.
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