Given G = (V, E) an undirected graph and two specified nonadjacent nodes a and b of V , a cut separator is a subset F = ...... such that a, b E V \ C and a and b belong to different connected components of the graph induced by V \ C. Given a nonnegative cost vector C E RE+, the optimal cut separator problem is to find a 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