It has long been known that Feedback Vertex Set can be solved in time 2 Opw log wq n Op1q on n-vertex graphs of treewidth w, but it was only recently that this running time was improved to 2 Opwq n Op1q , 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 P of graphs, the Bounded 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 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 Opwd 2 q n Op1q ,• if P contains a graph with an induced cycle of length ě 4, then the problem is not solvable in time 2 opw log wq n Op1q even for fixed d " , unless the ETH fails.We also study a similar problem, called Bounded 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 P contains all chordal graphs, then it cannot be solved in time f pwqn opwq for some function f , unless the ETH fails.Bounded-size components. Using a similar technique, we can obtain analogous results for a simpler problem, which we call Bounded P-Component Vertex Deletion, where we want to remove at most k vertices such that each connected component of the resulting graph has at most d vertices and belongs to P. If we have only the size constraint (i.e., P contains every graph), then this problem is known as Component Order Connectivity [9]. Let P be a class of graphs.Bounded P-Component Vertex Deletion Parameter: d, w Input: A graph G of treewidth at most w, and positive integers d and k. Question: Is there a set S of at most k vertices in G such that each connected component of G´S has at most d vertices and is in P?Drange, Dregi, and van 't Hof [9] studied the parameterized complexity of a weighted variant of the Component Order Connectivity problem; their results imply, in particular, that Component Order Connectivity can be solved in time 2 Opk log dq n, but is W r1s-hard parameterized by only k or d. The corresponding edge-deletion problem, parameterized by treewidth, was studied by Enright and Meeks [10]. For general classes P, we prove results that are analogous to those for Bounded P-Block Vertex Deletion. Theorem 1.3. Let P be a class of graphs that is hereditary, recognizable in polynomial time, and consists of only chordal graphs. Then Bounded P-Component Vertex Deletion can be solved in time 2 Opwd 2 q k 2 n on graphs with n vertices and treewidth w.Theorem 1.4. Let P be a hereditary class of graphs that is polynomial-time recognizable. I...
