  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 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.
