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

• Main Authors: Paesani, Giacomo, Paulusma, Daniël, Rzążewski, Paweł
• Format: Journal Article
• Language: English
• Published: 06.05.2021
• Subjects: Computer Science - Computational Complexity; Computer Science - Data Structures and Algorithms; Computer Science - Discrete Mathematics; Mathematics - Combinatorics
• DOI: 10.48550/arxiv.2105.02736

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., TCS 2018],$(sP_1+P_3)$ -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 ( $sP_3)$ -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.