Proof that the graph optimization problem is NP-hard I'm trying to prove that the following optimization problem is NP-hard:

Given a graph $G=(V,E)$, non-negative vertex weight functions $w(v)$ and $s(v)$, and a non-negative edge weight function $t(u,v)$, find a subset of vertices $S \in V$ that minimizes the function $C$:

$$C = \sum_{v \notin S}{w(v)} + \sum_{v \in S}{s(v)} + \sum_{(u,v) \in E:u \in S, v \notin S}{t(u,v)} $$

$$w(v) \geq 0, s(v) \geq 0, t(u, v) \geq 0$$

In other words, I'm trying to bipartition the graph and minimize the sum of vertex weights and edge weights across the cut, but the vertex weights are different depending on which "side" of the partition they are.

I've tried to reduce the max-cut problem to this by somehow minimizing the complement graph, also tried the sparsest cut, but the second weight function always seems to be problematic. Obviously, assuming that $s(v) = 0$ or $s(v) = w(v)$ makes the problem trivial, so there must be some solution that I'm missing. Or maybe the problem isn't NP-hard at all?

Any help is appreciated :)

Edit: restrict to non-negative weights
This problem is in P, it can be reduced to the Minimum cut problem.

The graph construction is as follows - Add a source and a sink vertex. For each vertex $i$, add an edge with cost $w(i)$ from source to $i$ and another edge of cost $s(i)$ from $i$ to sink. Also add edges from $i$ to $j$ of cost $t(i,j)$ for every pair of vertices $i$ and $j$. 

The cost of the cut from source to sink in this graph with set of vertices $S$ along with source in one side of the cut exactly corresponds to your function $C$