Cops & Robber on Periodic Temporal Graphs: Characterization and Improved Bounds

Abstract

We study the classical Cops and Robber game when the cops and the robber move on an infinite periodic sequence $$\mathcal {G}= (G_0, \dots, G_{p-1})^*$$ of graphs on the same set V of n vertices: in round t, the topology of $$\mathcal {G}$$ is $$G:i=(V,E_i)$$ where $$i\equiv t\pmod {p}$$. As in the traditional case of static graphs, the main concern is on the characterization of the class of periodic temporal graphs where k cops can capture the robber. Concentrating on the case of a single cop, we provide a characterization of copwin periodic temporal graphs. Based on this characterization, we design an algorithm for determining if a periodic temporal graph is copwin with time complexity $$O(p\ n^2 + n\ m)$$, where $$m=\sum _{i\in \mathbb {Z}_p} |E_i|$$, improving the existing $$O(p\ n^3)$$ bound. Let us stress that, when $$p=1$$ (i.e., in the static case), the complexity becomes $$O(n\ m)$$, improving the best existing $$O(n^3)$$ bound.