A Quartic Kernel for Pathwidth-One Vertex Deletion

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(k^{4}) vertices, and (iii) k leq k. We also give a fixed parameter tractable (FPT) algorithm for the problem that runs in O(7^{k} k cdot n^{2}) time.