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

## TL;DR

This paper establishes new polynomial-time algorithms for cycle transversal problems on specific graph classes and proves NP-completeness on others, advancing understanding of computational complexity in graph theory.

## Contribution

It introduces polynomial-time solutions for Feedback Vertex Set and Odd Cycle Transversal on $(sP_1+P_3)$-free graphs and cographs, and proves NP-completeness on certain other graph classes.

## Key findings

- Polynomial-time algorithms for $(sP_1+P_3)$-free graphs.
- Polynomial-time algorithms for cographs.
- NP-completeness on $(P_2+P_5,P_6)$-free graphs.

## Abstract

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.