Smaller kernels for two vertex deletion problems

Abstract

In this paper we consider two vertex deletion problems. In the Block Vertex Deletion problem, the input is a graph G and an integer k, and the goal is to decide whether there is a set of at most k vertices whose removal from G result in a block graph (a graph in which every biconnected component is a clique). In the Pathwidth One Vertex Deletion problem, the input is a graph G and an integer k, and the goal is to decide whether there is a set of at most k vertices whose removal from G result in a graph with pathwidth at most one. We give a kernel for Block Vertex Deletion with $O\left(k^3\right)$ vertices and a kernel for Pathwidth One Vertex Deletion with $O\left(k^2\right)$ vertices. Our results improve the previous $O\left(k^4\right)$-vertex kernel for Block Vertex Deletion (Agrawal et al., 2016 [1]) and the $O\left(k^3\right)$-vertex kernel for Pathwidth One Vertex Deletion (Cygan et al., 2012 [3]).