  We show that the Minimal Length-Bounded L-But problem can be computed in
linear time with respect to L and the tree-width of the input graph as
parameters. In this problem the task is to find a set of edges of a graph such
that after removal of this set, the shortest path between two prescribed
vertices is at least L long. We derive an FPT algorithm for a more general
multi-commodity length bounded cut problem when parameterized by the number of
terminals also.
  For the former problem we show a W[1]-hardness result when the
parameterization is done by the path-width only (instead of the tree-width) and
that this problem does not admit polynomial kernel when parameterized by
tree-width and L. We also derive an FPT algorithm for the Minimal
Length-Bounded Cut problem when parameterized by the tree-depth. Thus showing
an interesting paradigm for this problem and parameters tree-depth and
path-width.
