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 \({\mathcal O}(k^{4})\) vertices, and (iii) k′ ≤ k. We also give a fixed parameter tractable (FPT) algorithm for the problem that runs in \({\mathcal O}(7^{k}k\cdot n^{2})\) time.KeywordsParameterized ComplexityPolynomial KernelTree DecompositionLayout ProblemReduction RuleThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.