Parameterized Vertex Deletion Problems for Hereditary Graph Classes with a Block Property

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 such that each block of G−S has at most d vertices and is in P. We show that when P satisfies a natural hereditary property and is recognizable in polynomial time, Bounded P-Block Vertex Deletion can be solved in time 2O(k log d)nO(1). When P contains all split graphs, we show that this running time is essentially optimal unless the Exponential Time Hypothesis fails. On the other hand, if P consists of only complete graphs, or only cycle graphs and K2, then Bounded P-Block Vertex Deletion admits a cknO(1)-time algorithm for some constant c independent of d. We also show that Bounded P-Block Vertex Deletion admits a kernel with O(kd) vertices.