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 on n vertices. 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 leaves, and eventually returns back to the center. We present here 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 each 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 fully characterize MAXSTAREXP(k) and show a dichotomy in terms of its complexity: on one hand, we show that for both k=2 and k=3 , it can be efficiently solved in O(nlogn) time; on the other hand, we show that it is APX-complete, for every k≥4 (does not admit a PTAS, unless P = NP, but admits a polynomial-time 1.582-approximation algorithm). We also partially characterize STAREXP(k) in terms of complexity: we show that it can be efficiently solved in O(nlogn) time for k∈{2,3} (as a corollary of the solution to MAXSTAREXP(k) , for k∈{2,3} ), but is NP-complete, for every k≥6

Non-Strict Temporal Exploration

A temporal graph G=〈G1,...,GL〉is a sequence of graphs Gi⊆G, for some given underlying graph G of order n. We consider the non-strict variant of the Temporal Exploration problem, in which we are asked to decide if G admits a sequence W of consecutively crossed edges e∈G, such that W visits all vertices at least once and that each e∈W is crossed at a timestep t′∈[L] such that t′≥t, where t is the time step during which the previous edge was crossed. This variant of the problem is shown to be NP-complete. We also consider the hardness of approximating the exploration time for yes-instances in which our order-ninput graph satisfies certain assumptions that ensure exploration schedules always exist. The first is that each pair of vertices are contained in the same component at least once in every period of nsteps, whilst the second is that the temporal diameter of our input graphis bounded by a constantc. For the latter of these two assumptions we showO(n12−ε)-inapproximability and O(n1−ε)-inapproximability in thec= 2 andc≥3 cases, respectively. For graphs with temporal diameterc= 2, we also prove an O(√nlogn) upper bound on worst-case time required for exploration, as well as anΩ(√n) lower boun