Partitioning H-free graphs of bounded diameter

Brause, C.; Golovach, P. A.; 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. A., Martin, B., Paulusma, D., & Smith, S. (2021, December). Partitioning H-free graphs of bounded diameter. Presented at 32nd International Symposium on Algorithms and Computation (ISAAC 2021), Fukuoka, Japan

• Presentation Conference Type: Conference Paper (published)
• Conference Name: 32nd International Symposium on Algorithms and Computation (ISAAC 2021)
• Start Date: Dec 6, 2021
• End Date: Dec 8, 2021
• Acceptance Date: Sep 6, 2021
• Online Publication Date: Nov 30, 2021
• Publication Date: 2021
• Deposit Date: Oct 23, 2021
• Publicly Available Date: Jan 17, 2023
• Pages: 21:1-21:14
• Series Title: Leibniz International Proceedings in Informatics
• Series ISSN: 1868-8969
• DOI: https://doi.org/10.4230/lipics.isaac.2021.21
• Public URL: https://durham-repository.worktribe.com/output/1138258