A cop and robber game on edge-periodic temporal graphs

Abstract

We introduce a cops and robbers game with one cop and one robber on a special type of time-varying graphs (TVGs), namely edge-periodic graphs. These are TVGs in which, for each edge e, a binary string $\tau \left(e\right)$ is given such that the edge e is present in time step t if and only if $\tau \left(e\right)$ contains a 1 at position $t\mathrm{mod}\left|\tau \right(e\left)\right|$. This periodicity allows for a compact representation of infinite TVGs. We prove that even for very simple underlying graphs, i.e., directed and undirected cycles, the problem of deciding whether a cop-winning strategy exists is NP-hard and $W\left[1\right]$-hard parameterized by the number of vertices. Furthermore, we show that this decision problem can be solved on general edge-periodic graphs in PSPACE. Finally, we present tight bounds on the minimum length of a directed or undirected cycle that guarantees the cycle to be robber-winning.