  A temporal graph has an edge set that may change over discrete time steps,
and a temporal path (or walk) must traverse edges that appear at increasing
time steps. Accordingly, two temporal paths (or walks) are temporally disjoint
if they do not visit any vertex at the same time. The study of the
computational complexity of finding temporally disjoint paths or walks in
temporal graphs has recently been initiated by Klobas et al. [IJCAI '21]. This
problem is motivated by applications in multi-agent path finding (MAPF), which
include robotics, warehouse management, aircraft management, and traffic
routing.
  We extend Klobas et al.'s research by providing parameterized hardness
results for very restricted cases, with a focus on structural parameters of the
so-called underlying graph. On the positive side, we identify sufficiently
simple cases where we can solve the problem efficiently. Our results reveal
some surprising differences between the "path version" and the "walk version"
(where vertices may be visited multiple times) of the problem, and answer
several open questions posed by Klobas et al.
