An Improved FPT Algorithm and a Quadratic Kernel for Pathwidth One Vertex Deletion

Abstract

The Pathwidth One Vertex Deletion (POVD) problem asks whether, given an undirected graph G and an integer k, one can delete at most k vertices from G so that the remaining graph has pathwidth at most 1. The question can be considered as a natural variation of the extensively studied Feedback Vertex Set (FVS) problem, where the deletion of at most k vertices has to result in the remaining graph having treewidth at most 1 (i.e., being a forest). Recently Philip et al. (WG, Lecture Notes in Computer Science, vol. 6410, pp. 196–207, 2010) initiated the study of the parameterized complexity of POVD, showing a quartic kernel and an algorithm which runs in time 7k n O(1). In this article we improve these results by showing a quadratic kernel and an algorithm with time complexity 4.65k n O(1), thus obtaining almost tight kernelization bounds when compared to the general result of Dell and van Melkebeek (STOC, pp. 251–260, ACM, New York, 2010). Techniques used in the kernelization are based on the quadratic kernel for FVS, due to Thomasse (ACM Trans. Algorithms 6(2), 2010).