  Given a graph $G=(V,E)$ with two distinguished vertices $s,t\in V$ and an
integer parameter $L>0$, an {\em $L$-bounded cut} is a subset $F$ of edges
(vertices) such that the every path between $s$ and $t$ in $G\setminus F$ has
length more than $L$. The task is to find an $L$-bounded cut of minimum
cardinality.
  Though the problem is very simple to state and has been studied since the
beginning of the 70's, it is not much understood yet. The problem is known to
be $\cal{NP}$-hard to approximate within a small constant factor even for
$L\geq 4$ (for $L\geq 5$ for the vertex cuts). On the other hand, the best
known approximation algorithm for general graphs has approximation ratio only
$\mathcal{O}({n^{2/3}})$ in the edge case, and $\mathcal{O}({\sqrt{n}})$ in the
vertex case, where $n$ denotes the number of vertices.
  We show that for planar graphs, it is possible to solve both the edge- and
the vertex-version of the problem optimally in time $\mathcal{O}(L^{3L}n)$.
That is, the problem is fixed parameter tractable (FPT) with respect to $L$ on
planar graphs. Furthermore, we show that the problem remains FPT even for
bounded genus graphs, a super class of planar graphs.
  Our second contribution deals with approximations of the vertex version of
the problem. We describe an algorithm that for a given a graph $G$, its tree
decomposition of treewidth $\tau$ and vertices $s$ and $t$ computes a
$\tau$-approximation of the minimum $L$-bounded $s-t$ vertex cut; if the
decomposition is not given, then the approximation ratio is $\mathcal{O}(\tau
\sqrt{\log \tau})$. For graphs with treewidth bounded by
$\mathcal{O}(n^{1/2-\epsilon})$ for any $\epsilon>0$, but not by a constant,
this is the best approximation in terms of~$n$ that we are aware of.
