  In the presented paper we study the Length-Bounded Cut problem for special
graph classes as well as from a parameterized-complexity viewpoint. Here, we
are given a graph $G$, two vertices $s$ and $t$, and positive integers $\beta$
and $\lambda$. The task is to find a set of edges $F$ of size at most $\beta$
such that every $s$-$t$-path of length at most $\lambda$ in $G$ contains some
edge in $F$.
  Bazgan et al. conjectured that Length-Bounded Cut admits a polynomial-time
algorithm if the input graph $G$ is a~proper interval graph. We confirm this
conjecture by showing a dynamic-programming based polynomial-time algorithm. We
strengthen the W[1]-hardness result of Dvo\v{r}\'ak and Knop. Our reduction is
shorter, seems simpler to describe, and the target of the reduction has
stronger structural properties. Consequently, we give W[1]-hardness for the
combined parameter pathwidth and maximum degree of the input graph. Finally, we
prove that Length-Bounded Cut is W[1]-hard for the feedback vertex number. Both
our hardness results complement known XP algorithms.
