  We study the existence of polynomial kernels, for parameterized problems
without a polynomial kernel on general graphs, when restricted to graphs of
bounded twin-width. Our main result is that a polynomial kernel for
$k$-Dominating Set on graphs of twin-width at most 4 would contradict a
standard complexity-theoretic assumption. The reduction is quite involved,
especially to get the twin-width upper bound down to 4, and can be tweaked to
work for Connected $k$-Dominating Set and Total $k$-Dominating Set (albeit with
a worse upper bound on the twin-width). The $k$-Independent Set problem admits
the same lower bound by a much simpler argument, previously observed [ICALP
'21], which extends to $k$-Independent Dominating Set, $k$-Path, $k$-Induced
Path, $k$-Induced Matching, etc. On the positive side, we obtain a simple
quadratic vertex kernel for Connected $k$-Vertex Cover and Capacitated
$k$-Vertex Cover on graphs of bounded twin-width. Interestingly the kernel
applies to graphs of Vapnik-Chervonenkis density 1, and does not require a
witness sequence. We also present a more intricate $O(k^{1.5})$ vertex kernel
for Connected $k$-Vertex Cover. Finally we show that deciding if a graph has
twin-width at most 1 can be done in polynomial time, and observe that most
optimization/decision graph problems can be solved in polynomial time on graphs
of twin-width at most 1.
