A periodic temporal graph $\mathcal{G}=(G_0, G_1, \dots, G_{p-1})^*$ is an infinite periodic sequence of graphs $G_i=(V,E_i)$ where $G=(V,\cup_i E_i)$ is called the footprint. Recently, the arena where the Cops and Robber game is played has been extended from a graph to a periodic graph; in this case, the copnumber is also the minimum number of cops sufficient for capturing the robber. We study the connections and distinctions between the copnumber $c(\mathcal{G})$ of a periodic graph $\mathcal{G}$ and the copnumber $c(G)$ of its footprint $G$ and establish several facts. For instance, we show that the smallest periodic graph with $c(\mathcal{G}) = 3$ has at most $8$ nodes; in contrast, the smallest graph $G$ with $c(G) = 3$ has $10$ nodes. We push this investigation by generating multiple examples showing how the copnumbers of a periodic graph $\mathcal{G}$, the subgraphs $G_i$ and its footprint $G$ can be loosely tied. Based on these results, we derive upper bounds on the copnumber of a periodic graph from properties of its footprint such as its treewidth.

翻译：周期时间图 $\mathcal{G}=(G_0, G_1, \dots, G_{p-1})^*$ 是一个无限周期序列图 $G_i=(V,E_i)$，其中 $G=(V,\cup_i E_i)$ 称为其足迹。近年来，"警察与强盗"游戏的博弈空间已从单一图扩展到周期图；在此情形下，警察数仍是足以捕获强盗所需的最少警察数量。我们研究了周期图 $\mathcal{G}$ 的警察数 $c(\mathcal{G})$ 与其足迹 $G$ 的警察数 $c(G)$ 之间的联系与差异，并建立了若干结论。例如，我们证明了满足 $c(\mathcal{G}) = 3$ 的最小周期图最多包含 $8$ 个节点；相比之下，满足 $c(G) = 3$ 的最小图 $G$ 需 $10$ 个节点。我们通过生成多个示例进一步探究，揭示了周期图 $\mathcal{G}$、子图 $G_i$ 及其足迹 $G$ 的警察数之间可能存在松散关联。基于这些结果，我们从足迹的树宽等性质出发，推导了周期图警察数的上界。