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

Abstract

For a class of graphs $\mathcal {P}$, the Bounded $\mathcal {P}$-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 $\mathcal {P}$. We show that when $\mathcal {P}$ satisfies a natural hereditary property and is recognizable in polynomial time, Bounded $\mathcal {P}$-Block Vertex Deletion can be solved in time $2^{\mathcal {O}(k \log d)}n^{\mathcal {O}(1)}$, and this running time cannot be improved to $2^{o(k \log d)}n^{\mathcal {O}(1)}$, in general, unless the Exponential Time Hypothesis fails. On the other hand, if $\mathcal {P}$ consists of only complete graphs, or only $K_1, K_2$, and cycle graphs, then Bounded $\mathcal {P}$-Block Vertex Deletion admits a $c^{k}n^{\mathcal {O}(1)}$-time algorithm for some constant c independent of d. We also show that Bounded $\mathcal {P}$-Block Vertex Deletion admits a kernel with $\mathcal {O}(k^2 d^7)$ vertices.

All authors are supported by ERC Starting Grant PARAMTIGHT (No. 280152).