# Finding vertex separator such that the induced subgraph has minimal number of edges

My problem is related to edge and vertex cuts with a little twist.

Given a graph $$G$$ and two vertexes $$u$$ and $$v$$. I want to find a set of vertexes $$S \subset V$$ that disconnects $$u$$ and $$v$$ such that the induced subgraph $$G[S]$$ has minimal number of edges.

Consider the following graph:

The red edges are the minimal edge cut. The blue vertices are the minimal vertex cut. The green vertices are a vertex cut such that the induced subgraph has the least number of edges. I try to find the green cut. In fact, the number of vertices is not required to be minimal as long as the number of edges in $$G[S]$$ are minimal.

Is there an algorithm in $$P$$ to solve this problem? Do you have an idea how to approach the problem?