Faster parameterized algorithms for two vertex deletion problems

In the l-PATH VERTEX COVER problem (resp., the l-COMPONENT ORDER CONNECTIVITY problem) the input is an undirected graph G and an integer k. The goal is to decide whether there is a set of vertices of size at most k whose deletion from G results in a graph that does not contain a path with l vertices (resp., does not contain a connected component with at least l vertices). In this paper we give a parameterized algorithm for l-PATH VERTEX COVER when l=5,6,7, whose running times are O^⁎(3.945^k), O^⁎(4.947^k), and O^⁎(5.951^k), respectively. We also give an algorithm for l-COMPONENT ORDER CONNECTIVITY whose running time is O^⁎((l−1−ϵ_l)^k) for every l≥4, where ϵ_l>0 for every l.