# The temporal explorer who returns to the base

In this paper we study the problem of exploring a temporal graph (i.e. a graph that changes over time), in the fundamental case where the underlying static graph is a star. The aim of the exploration problem in a temporal star is to find a temporal walk which starts at the center of the star, visits all leafs, and eventually returns back to the center. We initiate a systematic study of the computational complexity of this problem, depending on the number k of time-labels that every edge is allowed to have; that is, on the number k of time points where every edge can be present in the graph. To do so, we distinguish between the decision version StarExp(k) asking whether a complete exploration of the instance exists, and the maximization version MaxStarExp(k) of the problem, asking for an exploration schedule of the greatest possible number of edges in the star. We present here a collection of results establishing the computational complexity of these two problems. On the one hand, we show that both MaxStarExp(2) and StarExp(3) can be efficiently solved in O(n log n) time on a temporal star with n vertices. On the other hand, we show that, for every k >= 6, StarExp(k) is NP-complete and MaxStarExp(k) is APX-hard, and thus it does not admit a PTAS, unless P = NP. The latter result is complemented by a polynomial-time 2-approximation algorithm for MaxStarExp(k), for every k, thus proving that MaxStarExp(k) is APX-complete. Finally, we give a partial characterization of the classes of temporal stars with random labels which are, asymptotically almost surely, yes-instances and no-instances for StarExp(k) respectively.

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