Before revision:  In particular, we show that for (3P_1+P_2)-free graphs (where P_r is the path on r vertices) there is a polynomial-time algorithm, but the problem is NP-complete for 5P_1-free graphs, that is, graphs of independence number~4. Thus we obtain a dichotomy with respect to the independence number; this is an analogue of the dichotomy for Weighted Subset Feedback Vertex Set recently obtained by Papadopoulos and Tzimas.
After revision:  Our general approach can also be used for Weighted Subset Feedback Vertex Set recently obtained by Papadopoulos and Tzimas.