Partitioning H-free graphs of bounded diameter

Brause, C.; Golovach, P.; Martin, B.; Paulusma, D.; Smith, S.

Abstract

A natural way of increasing our understanding of NP-complete graph problems is to restrict the input to a special graph class. Classes of H-free graphs, that is, graphs that do not contain some graph H as an induced subgraph, have proven to be an ideal testbed for such a complexity study. However, if the forbidden graph H contains a cycle or claw, then these problems often stay NP-complete. A recent complexity study (MFCS 2019) on the k-Colouring problem shows that we may still obtain tractable results if we also bound the diameter of the H-free input graph. We continue this line of research by initiating a complexity study on the impact of bounding the diameter for a variety of classical vertex partitioning problems restricted to H-free graphs. We prove that bounding the diameter does not help for Independent Set, but leads to new tractable cases for problems closely related to 3-Colouring. That is, we show that Near-Bipartiteness, Independent Feedback Vertex Set, Independent Odd Cycle Transversal, Acyclic 3-Colouring and Star 3-Colouring are all polynomial-time solvable for chair-free graphs of bounded diameter. To obtain these results we exploit a new structural property of 3-colourable chair-free graphs.

Citation

Brause, C., Golovach, P., Martin, B., Paulusma, D., & Smith, S. (2022). Partitioning H-free graphs of bounded diameter. Theoretical Computer Science, 930, 37-52. https://doi.org/10.1016/j.tcs.2022.07.009

• Journal Article Type: Article
• Acceptance Date: Jul 10, 2022
• Online Publication Date: Jul 16, 2022
• Publication Date: Sep 21, 2022
• Deposit Date: Oct 16, 2022
• Publicly Available Date: Jul 16, 2023
• Journal: Theoretical Computer Science
• Print ISSN: 0304-3975
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 930
• Pages: 37-52
• DOI: https://doi.org/10.1016/j.tcs.2022.07.009
• Public URL: https://durham-repository.worktribe.com/output/1188918