  The CONNECTED VERTEX COVER problem asks for a vertex cover in a graph that
induces a connected subgraph. The problem is known to be fixed-parameter
tractable (FPT), and is unlikely to have a polynomial sized kernel (under
complexity theoretic assumptions) when parameterized by the solution size. In a
recent paper, Lokshtanov et al.[STOC 2017], have shown an $\alpha$-approximate
kernel for the problem for every $\alpha > 1$, in the framework of approximate
or lossy kernelization. In this work, we exhibit lossy kernels and FPT
algorithms for CONNECTED VERTEX COVER for parameters that are more natural and
functions of the input, and in some cases, smaller than the solution size. The
parameters we consider are the sizes of a split deletion set, clique deletion
set, clique cover, cluster deletion set and chordal deletion set.
