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A graph is $H$-free if it contains no induced subgraph isomorphic to $H$. Weprove new complexity results for the two classical cycle transversal problemsFeedback Vertex Set and Odd Cycle Transversal by showing that they can besolved in polynomial time on $(sP_1+P_3)$-free graphs for every integer $s\geq1$. We show the same result for the variants Connected Feedback Vertex Set andConnected Odd Cycle Transversal. We also prove that the latter two problems arepolynomial-time solvable on cographs; this was already known for FeedbackVertex Set and Odd Cycle Transversal. We complement these results by provingthat Odd Cycle Transversal and Connected Odd Cycle Transversal are NP-completeon $(P_2+P_5,P_6)$-free graphs.

On Cycle Transversals and Their Connected Variants in the Absence of a Small Linear Forest