Partitioning H-Free Graphs of Bounded Diameter
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 -free graphs, that is, graphs that do not contain some graph as an induced subgraph, have proven to be an ideal testbed for such a complexity study. However, if the forbidden graph contains a cycle or claw, then these problems often stay NP-complete. A recent complexity study on the -Colouring problem shows that we may still obtain tractable results if we also bound the diameter of the -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 -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.
Cite
@article{arxiv.2105.04588,
title = {Partitioning H-Free Graphs of Bounded Diameter},
author = {Christoph Brause and Petr Golovach and Barnaby Martin and Daniël Paulusma and Siani Smith},
journal= {arXiv preprint arXiv:2105.04588},
year = {2022}
}