  The recently introduced graph parameter tree-cut width plays a similar role
with respect to immersions as the graph parameter treewidth plays with respect
to minors. In this paper, we provide the first algorithmic applications of
tree-cut width to hard combinatorial problems. Tree-cut width is known to be
lower-bounded by a function of treewidth, but it can be much larger and hence
has the potential to facilitate the efficient solution of problems that are not
known to be fixed-parameter tractable (FPT) when parameterized by treewidth. We
introduce the notion of nice tree-cut decompositions and provide FPT algorithms
for the showcase problems Capacitated Vertex Cover, Capacitated Dominating Set,
and Imbalance parameterized by the tree-cut width of an input graph. On the
other hand, we show that List Coloring, Precoloring Extension, and Boolean CSP
(the latter parameterized by the tree-cut width of the incidence graph) are
W[1]-hard and hence unlikely to be fixed-parameter tractable when parameterized
by tree-cut width.
