Feedback Vertex Set and Even Cycle Transversal for H-free graphs: Finding large block graphs

Paesani, G.; Paulusma, D.l; Rzążwski, P.

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 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., Theoret. Comput. Sci., 705 (2018), pp. 75--83], (sP1 +P3)-free graphs [Dabrowski et al., Algorithmica, 82 (2020), pp. 2841--2866] and P5-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 (sP3)-free or (sP1 + P5)-free graph G a largest induced subgraph whose blocks belong to some finite class \scrC 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.

Citation

Paesani, G., Paulusma, D., & Rzążwski, P. (2022). Feedback Vertex Set and Even Cycle Transversal for H-free graphs: Finding large block graphs. SIAM Journal on Discrete Mathematics, 36(4), 2453-2472. https://doi.org/10.1137/22m1468864

• Journal Article Type: Article
• Acceptance Date: Jun 1, 2022
• Online Publication Date: Oct 4, 2022
• Publication Date: 2022
• Deposit Date: Oct 16, 2022
• Publicly Available Date: Oct 18, 2022
• Journal: SIAM Journal on Discrete Mathematics
• Print ISSN: 0895-4801
• Electronic ISSN: 1095-7146
• Publisher: Society for Industrial and Applied Mathematics
• Peer Reviewed: Peer Reviewed
• Volume: 36
• Issue: 4
• Pages: 2453-2472
• DOI: https://doi.org/10.1137/22m1468864
• Public URL: https://durham-repository.worktribe.com/output/1191527