  A stable cutset in a graph $G$ is a set $S\subseteq V(G)$ such that vertices
of $S$ are pairwise non-adjacent and such that $G-S$ is disconnected, i.e., it
is both stable (or independent) set and a cutset (or separator). Unlike general
cutsets, it is $NP$-complete to determine whether a given graph $G$ has any
stable cutset. Recently, Rauch et al.\ [FCT 2023] gave a number of
fixed-parameter tractable (FPT) algorithms, time $f(k)\cdot |V(G)|^c$, for
Stable Cutset under a variety of parameters $k$ such as the size of a (given)
dominating set, the size of an odd cycle transversal, or the deletion distance
to $P_5$-free graphs. Earlier works imply FPT algorithms relative to
clique-width and relative to solution size.
  We complement these findings by giving the first results on the existence of
polynomial kernelizations for \stablecutset, i.e., efficient preprocessing
algorithms that return an equivalent instance of size polynomial in the
parameter value. Under the standard assumption that $NP\nsubseteq coNP/poly$,
we show that no polynomial kernelization is possible relative to the deletion
distance to a single path, generalizing deletion distance to various graph
classes, nor by the size of a (given) dominating set. We also show that under
the same assumption no polynomial kernelization is possible relative to
solution size, i.e., given $(G,k)$ answering whether there is a stable cutset
of size at most $k$. On the positive side, we show polynomial kernelizations
for parameterization by modulators to a single clique, to a cluster or a
co-cluster graph, and by twin cover.
