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

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 ( s P 1 + P 3 ) -free graphs for every integer s ≥ 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.