Parameterized temporal exploration problems

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.