A quartic kernel for pathwidth-one vertex deletion

Abstract

The pathwidth of a graph is a measure of how path-like the graph is. Given a graph G and an integer k, the problem of finding whether there exist at most k vertices in G whose deletion results in a graph of pathwidth at most one is NP-Complete. We initiate the study of the parameterized complexity of this problem, parameterized by k. We show that the problem has a quartic vertex-kernel: We show that, given an input instance (G = (V,E),k);|V| = n, we can construct, in polynomial time, an instance (G′,k′) such that (i) (G,k) is a YES instance if and only if (G′,k′) is a YES instance, (ii) G′ has O(k4)vertices, and (iii) k′ ≤ k. We also give a fixed parameter tractable (FPT) algorithm for the problem that runs in O(7kk·n2) time. © 2010 Springer-Verlag.