  An enumeration kernel as defined by Creignou et al. [Theory Comput. Syst.
2017] for a parameterized enumeration problem consists of an algorithm that
transforms each instance into one whose size is bounded by the parameter plus a
solution-lifting algorithm that efficiently enumerates all solutions from the
set of the solutions of the kernel. We propose to consider two new versions of
enumeration kernels by asking that the solutions of the original instance can
be enumerated in polynomial time or with polynomial delay from the kernel
solutions. Using the NP-hard Matching Cut problem parameterized by structural
parameters such as the vertex cover number or the cyclomatic number of the
input graph, we show that the new enumeration kernels present a useful notion
of data reduction for enumeration problems which allows to compactly represent
the set of feasible solutions.
