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

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 $s\geq 1$, both problems are polynomial-time solvable for $sP3$-free graphs and $(sP1+P5)$-free graphs; here, the graph $sP3$ denotes the disjoint union of $s$ paths on three vertices and the graph $sP1+P5$ 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 $sP2$-free graphs [Chiarelli et al., TCS 2018], $(sP1+P3)$-free graphs [Dabrowski et al., Algorithmica 2020] and $P5$-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 ($sP3)$-free or $(sP1+P_5)$-free graph $G$ a largest induced subgraph whose blocks belong to some finite class ${\cal C}$ 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.