  We consider the cops and robber game variant consisting of one cop and one
robber on time-varying graphs (TVG). The considered TVGs are edge periodic
graphs, i.e., for each edge, a binary string $s_e$ determines in which time
step the edge is present, namely the edge $e$ is present in time step $t$ if
and only if the string $s_e$ contains a $1$ at position $t \mod |s_e|$. This
periodicity allows for a compact representation of the infinite TVG. We proof
that even for very simple underlying graphs, i.e., directed and undirected
cycles the problem whether a cop-winning strategy exists is NP-hard and
W[1]-hard parameterized by the number of vertices. Our second main result are
matching lower bounds for the ration between the length of the underlying cycle
and the least common multiple (LCM) of the lengths of binary strings describing
edge-periodicies over which the graph is robber-winning. Our third main result
improves the previously known EXPTIME upper bound for Periodic Cop and Robber
on general edge periodic graphs to PSPACE-membership.
