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 do not contain some fixed graph $H$ as an induced subgraph. In particular, we every $s\geq 1$, both problems are polynomial-time solvable $sP_3$-free $(sP_1+P_5)$-free graphs; here, the $sP_3$ denotes disjoint union of $s$ paths three vertices $sP_1+P_5$ isolated a path five vertices. Our extend all known namely $sP_2$-free [Chiarelli et al., Theoret. Comput. Sci., 705 (2018), pp. 75--83], $(sP_1+P_3)$-free [Dabrowski Algorithmica, 82 (2020), 2841--2866] $P_5$-free [Abrishami Induced subgraphs bounded treewidth container method, in Proceedings 2021 ACM-SIAM Symposium Discrete Algorithms (SODA), SIAM, Philadelphia, 2021, 1948--1964]. Together, also show exhibit same behavior (subject to open cases). This is part due general algorithm design finding ($sP_3)$-free or $G$ largest subgraph whose blocks belong finite class ${\cal C}$ graphs. compare our with state-of-the-art Odd problem, which behave differently