Title: 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 $s\geq 1$, both problems are polynomial-time solvable for $sP_3$-free graphs and $(sP_1+P_5)$-free graphs; here, the graph $sP_3$ denotes the disjoint union of $s$ paths on three vertices and the graph $sP_1+P_5$ 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 $sP_2$-free graphs [Chiarelli et al., Theoret. Comput. Sci., 705 (2018), pp. 75--83], $(sP_1+P_3)$-free graphs [Dabrowski et al., Algorithmica, 82 (2020), pp. 2841--2866] and $P_5$-free graphs [Abrishami et al., Induced subgraphs of bounded treewidth and the container method, in Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), SIAM, Philadelphia, 2021, pp. 1948--1964]. Together, the new results also show that both problems exhibit the same behavior 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 ($sP_3)$-free or $(sP_1+P_5)$-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.