  A graph is $H$-free if it contains no induced subgraph isomorphic to $H$. We
prove new complexity results for the two classical cycle transversal problems
Feedback Vertex Set and Odd Cycle Transversal by showing that they can be
solved in polynomial time on $(sP_1+P_3)$-free graphs for every integer $s\geq
1$. We show the same result for the variants Connected Feedback Vertex Set and
Connected Odd Cycle Transversal. We also prove that the latter two problems are
polynomial-time solvable on cographs; this was already known for Feedback
Vertex Set and Odd Cycle Transversal. We complement these results by proving
that Odd Cycle Transversal and Connected Odd Cycle Transversal are NP-complete
on $(P_2+P_5,P_6)$-free graphs.
