  The (\textsc{Weighted}) \textsc{Subset Feedback Vertex Set} problem is a
generalization of the classical \textsc{Feedback Vertex Set} problem and asks
for a vertex set of minimum (weighted) size that intersects all cycles
containing a vertex of a predescribed set of vertices. Although the two
problems exhibit different computational complexity on split graphs, no similar
characterization is known on other classes of graphs. Towards the understanding
of the complexity difference between the two problems, it is natural to study
the importance of a structural graph parameter. Here we consider graphs of
bounded independent set number for which it is known that \textsc{Weighted
Feedback Vertex Set} is solved in polynomial time. We provide a dichotomy
result with respect to the size of a maximum independent set. In particular we
show that \textsc{Weighted Subset Feedback Vertex Set} can be solved in
polynomial time for graphs of independent set number at most three, whereas we
prove that the problem remains NP-hard for graphs of independent set number
four. Moreover, we show that the (unweighted) \textsc{Subset Feedback Vertex
Set} problem can be solved in polynomial time on graphs of bounded independent
set number by giving an algorithm with running time $n^{O(d)}$, where $d$ is
the size of a maximum independent set of the input graph. To complement our
results, we demonstrate how our ideas can be extended to other terminal set
problems on graphs of bounded independent set size. Based on our findings for
\textsc{Subset Feedback Vertex Set}, we settle the complexity of \textsc{Node
Multiway Cut}, a terminal set problem that asks for a vertex set of minimum
size that intersects all paths connecting any two terminals, as well as its
variants where nodes are weighted and/or the terminals are deletable, for every
value of the given independent set number.
