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 (sP1 + P3)-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 (P2 +P5, P6)-free graphs.called Connected Vertex Cover, Connected Feedback Vertex Set and Connected Odd Cycle Transversal, respectively. Garey and Johnson [15] proved that Connected Vertex Cover is NP-complete even on planar graphs of maximum degree 4 (see, for example, [14,31,36] for NPcompleteness results for other graph classes). Grigoriev and Sitters [18] proved that Connected Feedback Vertex Set is NP-complete even on planar graphs with maximum degree 9. More recently, Chiarelli et al. [10] proved that Connected Odd Cycle Transversal is NP-complete even on graphs of arbitrarily large girth and on line graphs.As all three decision problems and their connected variants are NP-complete, we can consider how to restrict the input to some special graph class in order to achieve tractability. Note that this approach is in line with the aforementioned results in the literature, where NP-completeness was proven on special graph classes. It is also in line with with, for instance, polynomial-time results for Connected Vertex Cover by Escoffier, Gourvès and Monnot [12] (for chordal graphs) and Ueno, Kajitani and Gotoh [35] (for graphs of maximum degree at most 3 and trees).Just as in most of these papers, we consider hereditary graph classes, that is, graph classes closed under vertex deletion. Hereditary graph classes form a rich framework that captures many well-studied graph classes. It is not difficult to see that every hereditary graph class G can be characterized by a (possibly infinite) set F G of forbidden induced subgraphs. If |F G | = 1, say F = {H}, then G is said to be monogenic, and every graph G ∈ G is said to be H-free. Considering monogenic graph classes can be seen as a natural first step for increasing our knowledge of the complexity of an NP-complete problem in a systematic way. Hence, we consider the following research question:How does the structure of a graph H influence the computational complexity of a graph transversal problem for input graphs that are H-free?Note that different graph transversal problems may behave differently on some class of H-free graphs. However, the general strategy for obtaining complexity results is to first try to prove that the restriction to H-free graphs is NP-complete whenever H contains a cycle or the claw (the 4-vertex star). This is usually done by showing, respectively, that the probl...
