Connectivity problems like k-Path and k-Disjoint Paths relate to many important milestones in parameterized complexity, namely the Graph Minors Project, color coding, and the recent development of techniques for obtaining kernelization lower bounds. This work explores the existence of polynomial kernels for various path and cycle problems, by considering nonstandard parameterizations. We show polynomial kernels when the parameters are a given vertex cover, a modulator to a cluster graph, or a (promised) max leaf number. We obtain lower bounds via cross-composition, e.g., for Hamiltonian Cycle and related problems when parameterized by a modulator to an outerplanar graph.
Focusing on structural parameters, Bodlaender et al. REF obtained polynomial kernels with respect to the vertex cover number, vertex-deletion distance to a cluster graph and the max leaf number.
Kernel Bounds for Path and Cycle Problems
ArXiv
ArXiv
Mathematics Computer Science