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Suppose that a train is running along a railway network, starting from adesignated origin, with the goal of reaching a designated destination. Thenetwork, however, is of a special nature: every time the train traverses aswitch, the switch will change its position immediately afterwards. Hence, thenext time the train traverses the same switch, the other direction will betaken, so that directions alternate with each traversal of the switch. Given anetwork with origin and destination, what is the complexity of deciding whetherthe train, starting at the origin, will eventually reach the destination? It iseasy to see that this problem can be solved in exponential time, but we are notaware of any polynomial-time method. In this short paper, we prove that theproblem is in NP $\cap$ coNP. This raises the question whether we have justfailed to find a (simple) polynomial-time solution, or whether the complexitystatus is more subtle, as for some other well-known (two-player) graph games.

ARRIVAL: A zero-player graph game in NP $\cap$ coNP