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Temporal graphs are a popular modelling mechanism for dynamic complex systemsthat extend ordinary graphs with discrete time. Simply put, time progresses oneunit per step and the availability of edges can change with time. We considerthe complexity of solving $\omega$-regular games played on temporal graphswhere the edge availability is ultimately periodic and fixed a priori.  We show that solving parity games on temporal graphs is decidable in PSPACE,only assuming the edge predicate itself is in PSPACE. A matching lower boundalready holds for what we call punctual reachability games on static graphs,where one player wants to reach the target at a given, binary encoded, point intime. We further study syntactic restrictions that imply more efficientprocedures. In particular, if the edge predicate is in $P$ and is monotonicallyincreasing for one player and decreasing for the other, then the complexity ofsolving games is only polynomially increased compared to static graphs.

Parity Games on Temporal Graphs