  The perfect matching polytope, i.e. the convex hull of (incidence vectors of)
perfect matchings of a graph is used in many combinatorial algorithms. Kotzig,
Lov\'asz and Plummer developed a decomposition theory for graphs with perfect
matchings and their corresponding polytopes known as the tight cut
decomposition which breaks down every graph into a number of indecomposable
graphs, so called bricks. For many properties that are of interest on graphs
with perfect matchings, including the description of the perfect matching
polytope, it suffices to consider these bricks. A key result by Lov\'asz on the
tight cut decomposition is that the list of bricks obtained is the same
independent of the choice of tight cuts made during the tight cut decomposition
procedure. This implies that finding a tight cut decomposition is polynomial
time equivalent to finding a single tight cut.
  We generalise the notions of a tight cut, a tight cut contraction and a tight
cut decomposition to hypergraphs. By providing an example, we show that the
outcome of the tight cut decomposition on general hypergraphs is no longer
unique. However, we are able to prove that the uniqueness of the tight cut
decomposition is preserved on a slight generalisation of uniform hypergraphs.
Moreover, we show how the tight cut decomposition leads to a decomposition of
the perfect matching polytope of uniformable hypergraphs and that the
recognition problem for tight cuts in uniformable hypergraphs is polynomial
time solvable.
