If only a subset of nodes is to be transfered, and that the cardinality of this subset is undefined, then I agree with @LocalSolver. Otherwise (if all nodes have to be transfered $1$ by $1$), I believe the problem is not NP-hard (nor NP-complete):
Consider the following graph :
- Create a first layer with the $n$ nodes.
- Create a second layer with $n \times n$ nodes : $1.1, 1.2, ...,1.n,...,n.1,n.2, n.n$. Link node $i$ to $i.1,...,i.n$ forall $i=1,...,n$.
- Create a third layer with $n$ nodes. Link nodes $i.j$ from layer $2$ to node $j$ from layer $3$, forall $i=1,...,n$, $j=1,...,n$.
- Create a source node linked to layer $1$ and a sink node linked to layer $3$. Each of these edges have a flow that must equal exactly $1$.
The graph is illustrated bellow:
Now, a flow in this graph is a feasible solution of your problem (if my understanding is correct). Layer $2$ will give you the order of the transfers. For example, if edge $(1,1.3)$ is used, node $1$ will be transfered in 3rd position. The third layer ensures there is exactly $1$ node that is transfered per iteration.
To have the cheapest solution, add the cost of the cut on the edges linking layers $1$ and $2$.
Therefore your problem is not NP-complete, nor NP-hard, as flow problems can be solved in polynomial time.