The Travelling Salesman Problem literature is a good place to start.
1. Hardness Result The travelling tourist problem is NP-complete.
2. Changing origin Changing the starting city or the starting time changes the duration of the optimal tour by less than one day.
3. Representing a tour with a permutation There exists an optimal tour T, and a permutation P = p1, p2, . . .pn such that the "first-visits" occur in the order p1, . . ., pn, and the path between pi and p(i+1) is a fastest path.
4. Fastest Path Problem The fastest path problem is to find the minimum duration trip (given the start time) between a pair of cities. This can be solved with a version of Dijkstra's Algorithms.
5. Approximation Algorithms, Negative result If P != NP and r >= 1 there is no polynomial time approximation algorithm with ratio bound r for the travelling tourist problem.