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We analyze several time dependency issues for the selective traveling salesman problem with time-dependent profits. Specifically, we consider the case in which the profit collected at a vertex depends on the service time, understood as the time spent at this vertex, and when the service time at each vertex depends on the arrival time at the vertex. For each of these two cases, we formulate two continuous-time problems: (i) a vertex can be visited at most once, and (ii) vertices may be visited more than once. In each case, we consider general profit functions at the vertices, i.e., the profit functions are not limited to monotonic functions of time. We also formulate the problems as discrete-time problems using appropriate variants of an auxiliary time-extended graph, and we solve them with Gurobi. We apply our methodology to two sets of instances. First, we use a set of artificial instances to illustrate the main differences …