The temporal explorer who returns to the base

We study here the problem of exploring a temporal graph when the underlying graph is a star. The aim of the exploration problem in a temporal star is finding a temporal walk which starts and finishes at the center of the star, and visits all leaves. We present a systematic study of the computational complexity of this problem, depending on the number k of time points where each edge can be present in the graph. We distinguish between the decision version STAREXP(k), asking whether a complete exploration exists, and the maximization version MAXSTAREXP(k), asking for an exploration of the greatest possible number of edges. We fully characterize MAXSTAREXP(k) in terms of complexity. We also partially characterize STAREXP(k), showing that it is in P for k < 4, but is NP-complete, for every k > 5. Finally, we partially characterize classes of "random" temporal stars which are, asymptotically almost surely, yes-instances and no-instances for STAREXP(k). (C) 2021 Elsevier Inc. All rights reserved.