  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.
