  We prove that all quiver Grassmannians for exceptional representations of a
generalized Kronecker quiver admit a cell decomposition. In the process, we
introduce a class of regular representations which arise as quotients of
consecutive preprojective representations. Cell decompositions for quiver
Grassmannians of these "truncated preprojectives" are also established. We also
provide two natural combinatorial labelings for these cells. On the one hand,
they are labeled by certain subsets of a so-called 2-quiver attached to a
(truncated) preprojective representation. On the other hand, the cells are in
bijection with compatible pairs in a maximal Dyck path as predicted by the
theory of cluster algebras. The natural bijection between these two labelings
gives a geometric explanation for the appearance of Dyck path combinatorics in
the theory of quiver Grassmannians.
