**Abstract** : It has long been known that Feedback Vertex Set can be solved in time 2 O(w log w) n O(1) on graphs of treewidth w, but it was only recently that this running time was improved to 2 O(w) n 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 of graphs P, the Bounded P-Block Vertex Deletion problem asks, given a graph G on n vertices and positive integers k and d, whether there is a set S of at most k vertices of G such that each block of G − S has at most d vertices and is in P. Assuming that 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 P consists only of chordal graphs, then the problem can be solved in time 2 O(wd 2) n O(1) , if P contains a graph with an induced cycle of length 4, then the problem is not solvable in time 2 o(w log w) n O(1) even for fixed d = , unless the ETH fails. As a warm up, we consider the analogous Bounded P-Component Vertex Deletion problem requiring each connected component to be a member of P, and show that chordality is also the key to single-exponential parameterized algorithms for this problem.