Feedback Vertex Set and Even Cycle Transversal for H-Free Graphs: Finding Large Block Graphs

Abstract

We prove new complexity results for Feedback Vertex Set and Even Cycle Transversal on $H$-free graphs, that is, graphs that do not contain some fixed graph $H$ as an induced subgraph. In particular, we prove that for every $s\ge 1$, both problems are polynomial-time solvable for $s{P}_3$-free graphs and $(s{P}_1+P_5)$-free graphs; here, the graph $s{P}_3$ denotes the disjoint union of $s$ paths on three vertices and the graph $s{P}_1+P_5$ denotes the disjoint union of $s$ isolated vertices and a path on five vertices. Our new results for Feedback Vertex Set extend all known polynomial-time results for Feedback Vertex Set on $H$-free graphs, namely for $s{P}_2$-free graphs [Chiarelli et al., TCS 2018], $(s{P}_1+P_3)$-free graphs [Dabrowski et al., Algorithmica 2020] and $P_5$-free graphs [Abrishami et al., SODA 2021]. Together, the new results also show that both problems exhibit the same behaviour on $H$-free graphs (subject to some open cases). This is in part due to a new general algorithm we design for finding in a ($s{P}_3)$-free or $(s{P}_1+P_5)$-free graph $G$ a largest induced subgraph whose blocks belong to some finite class $\mathcal{C}$ of graphs. We also compare our results with the state-of-the-art results for the Odd Cycle Transversal problem, which is known to behave differently on $H$-free graphs.