We perform a systematic study in the computational complexity of the connected variant of three related transversal problems: Vertex Cover, Feedback Vertex Set, and Odd Cycle Transversal. Just like their original counterparts, these variants are NP-complete for general graphs. It is known that Connected Vertex Cover is NP-complete even for $H$-free graphs whenever $H$ contains a claw or a cycle. We show that the two other connected variants also remain NP-complete in such cases. In the remaining case $H$ is a disjoint union of paths. We show that Connected Vertex Cover, Connected Feedback Vertex Set, and Connected Odd Cycle Transversal are polynomial-time solvable for $sP_2$-free graphs for every constant $s\geq 1$.

For proving these results we use polynomial-time algorithms for enumerating all minimal vertex covers, minimal feedback vertex sets, and minimal odd cycle transversals of a given $sP_2$-free graph. We also use known results on the price of connectivity for vertex covers, feedback vertex sets, and odd cycle transversals. These are the first applications of the price of connectivity that result in polynomial-time algorithms.

This is joint work with N. Chiarelli, T. R. Hartinger, M. Johnson, and D. Paulusma. This work was supported by a London Mathematical Society Scheme 4 Grant, Leverhulme Trust Grant RPG-2016-258 and by the Slovenian Research Agency (I0-0035, research programs P1-0285, research projects N1-0032, J1-5433, J1-6720, J1-6743, J1-7051, and a Young Researchers Grant).