  We present a method for reducing the treewidth of a graph while preserving
all the minimal $s-t$ separators. This technique turns out to be very useful
for establishing the fixed-parameter tractability of constrained separation and
bipartization problems. To demonstrate the power of this technique, we prove
the fixed-parameter tractability of a number of well-known separation and
bipartization problems with various additional restrictions (e.g., the vertices
being removed from the graph form an independent set). These results answer a
number of open questions in the area of parameterized complexity.
