  In this paper we study the fixed-parameter tractability of the problem of
deciding whether a given temporal graph admits a temporal walk that visits all
vertices (temporal exploration) or, in some problem variants, a certain subset
of the vertices. Formally, a temporal graph is a sequence <G_1,...,G_L> of
graphs with V(G_t) = V(G) and E(G_t) a subset of E(G) for all t in [L] and some
underlying graph G, and a temporal walk is a time-respecting sequence of
edge-traversals. We consider both the strict variant, in which edges must be
traversed in strictly increasing timesteps, and the non-strict variant, in
which an arbitrary number of edges can be traversed in each timestep. For both
variants, we give FPT algorithms for the problem of finding a temporal walk
that visits a given set X of vertices, parameterized by |X|, and for the
problem of finding a temporal walk that visits at least k distinct vertices in
V(G), parameterized by k. We also show W[2]-hardness for a set version of the
temporal exploration problem for both variants. For the non-strict variant, we
give an FPT algorithm for the temporal exploration problem parameterized by the
lifetime of the input graph, and we show that the temporal exploration problem
can be solved in polynomial time if the graph in each timestep has at most two
connected components.
