Counting the number of distinct s-t cuts in a oriented graph I am trying to find the number of distinct s-t cuts in a oriented unweighed graph. In an article Enumeration in Graphs p. 45 I found good way how to enumerate those cuts (section 7.3). Is there a faster or simpler algorithm I can use if I am interested only in the number of such cuts and I do not actually need to enumerate it?

The definition of a s-t cut I am using is following. We have a directed graph where two vertices are labelled S and T. Cut is a minimal set of edges of a graph such that by removing those edges there will no longer exist a path from vertex S to vertex T.

I tried to ask this at Stack Overflow and I have been pointed here.
Counting the number of s,t-cuts is #P-complete.  This is a result by Provan and Ball.


J. Scott Provan, Michael O. Ball: The Complexity of Counting Cuts and of Computing the Probability that a Graph is Connected. SIAM J. Comput. 12(4): 777-788 (1983)


Therefore, unless some complexity-theoretic collapse happens, you cannot get essentially faster algorithm than listing all of them.