Given G=(V,E) an undirected graph and two specified nonadjacent nodes a and b of V, a cut separator is a subset F=δ(C)subset of or equal toE such that a,bset membership, variantV C and a and b belong to different connected components of the graph induced by V C. Given a nonnegative cost vector View the MathML source, cut separator of minimum cost. This new problem is closely related to the vertex separator problem. In this paper, we give a polynomial time algorithm for this problem. We also present four equivalent linear formulations, and we show their tightness. Using these results we obtain an explicit short polyhedral description of the dominant of the cut separator polytope.