  Given a vertex-weighted graph $G=(V,E)$ and a set $S \subseteq V$, a subset
feedback vertex set $X$ is a set of the vertices of $G$ such that the graph
induced by $V \setminus X$ has no cycle containing a vertex of $S$. The
\textsc{Subset Feedback Vertex Set} problem takes as input $G$ and $S$ and asks
for the subset feedback vertex set of minimum total weight. In contrast to the
classical \textsc{Feedback Vertex Set} problem which is obtained from the
\textsc{Subset Feedback Vertex Set} problem for $S=V$, restricted to graph
classes the \textsc{Subset Feedback Vertex Set} problem is known to be
NP-complete on split graphs and, consequently, on chordal graphs. However as
\textsc{Feedback Vertex Set} is polynomially solvable for AT-free graphs, no
such result is known for the \textsc{Subset Feedback Vertex Set} problem on any
subclass of AT-free graphs. Here we give the first polynomial-time algorithms
for the problem on two unrelated subclasses of AT-free graphs: interval graphs
and permutation graphs. As a byproduct we show that there exists a
polynomial-time algorithm for circular-arc graphs by suitably applying our
algorithm for interval graphs. Moreover towards the unknown complexity of the
problem for AT-free graphs, we give a polynomial-time algorithm for
co-bipartite graphs. Thus we contribute to the first positive results of the
\textsc{Subset Feedback Vertex Set} problem when restricted to graph classes
for which \textsc{Feedback Vertex Set} is solved in polynomial time.
