# Fixed-parameter tractability of counting small minimum (S,T)-cuts

The parameterized complexity of counting minimum cuts stands as a natural question because Ball and Provan showed its #P-completeness. For any undirected graph G=(V,E) and two disjoint sets of its vertices S,T, we design a fixed-parameter tractable algorithm which counts minimum edge (S,T)-cuts parameterized by their size p. Our algorithm operates on a transformed graph instance. This transformation, called drainage, reveals a collection of at most n=| V | successive minimum (S,T)-cuts Z_i. We prove that any minimum (S,T)-cut X contains edges of at least one cut Z_i. This observation, together with Menger's theorem, allows us to build the algorithm counting all minimum (S,T)-cuts with running time 2^O(p^2)n^O(1). Initially dedicated to counting minimum cuts, it can be modified to obtain an FPT sampling of minimum edge (S,T)-cuts.

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