# Min Cut with Vertices

I have an undirected graph G with a set of vertices and edges. Each vertex has a weight w. Let's assume we have all vertices connected with some paths. I'm looking for a variant of the min-cut problem. I search for a subdivision of the graph into two parts which are connected by a group of vertices which weights sum up to a minimum value. Can you give me a pointer or a name to the problem (if it exists already)?