Partitioning H-free graphs of bounded diameter

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.