  The $d$-Cut problem is to decide if a graph has an edge cut such that each
vertex has at most $d$ neighbours at the opposite side of the cut. If $d=1$, we
obtain the intensively studied Matching Cut problem. The $d$-Cut problem has
been studied as well, but a systematic study for special graph classes was
lacking. We initiate such a study and consider classes of bounded diameter,
bounded radius and $H$-free graphs. We prove that for all $d\geq 2$, $d$-Cut is
polynomial-time solvable for graphs of diameter $2$, $(P_3+P_4)$-free graphs
and $P_5$-free graphs. These results extend known results for $d=1$. However,
we also prove several NP-hardness results for $d$-Cut that contrast known
polynomial-time results for $d=1$. Our results lead to full dichotomies for
bounded diameter and bounded radius and to almost-complete dichotomies for
$H$-free graphs.
