Finding small node sets that can not be avoided on paths from source to sink
Finding the smallest number of edges instead of vertices is the famous Minimum Cut problem which can be solved using any of the Max-Flow-Min-Cut algorithms. Picking vertices is a specific case of the generalised max-flow-min-cut theorem and similar algorithms can be used to solve it. In any case, it can be formulated as a linear program with polynomial complexity, which can be solved in polynomial time using any of the LP-solvers.