Cops & Robber on Periodic Temporal Graphs

TL;DR

It is proved that all the fundamental properties established for $k=1$ continue to hold, providing a characterization of $k-copwin periodic graphs, as well as a general strategy to determine if a periodic graph is $k$-copwin.

Abstract

We consider the Cops and Robber pursuit-evasion game when the edge-set of the graph is allowed to change in time, possibly at every round. Specifically, the game is played 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 $i\equiv t \pmod{p}$. Concentrating on the case of a single cop, we provide a characterization of copwin periodic temporal graphs, establishing several basic properties on their nature, and extending to the temporal domain classical C&R concepts such as covers and corners. Based on these results, we design an efficient algorithm for determining if a periodic temporal graph is copwin. We also consider the case of $k>1$ cops. By shifting from a representation in terms of directed graphs to one in terms of directed multi-hypergraphs, we prove that all the fundamental properties established for $k=1$ continue to hold, providing a characterization of $k$-copwin periodic graphs, as well as a general strategy to determine if a periodic graph is $k$-copwin. Our results do not rely on any assumption on properties such as connectivity, symmetry, reflexivity held by the individual graphs in the sequence. They are established for a unified version of the game that includes the standard games studied in the literature, both for undirected and directed graphs, and both when the players are fully active and when they are not. They hold also for a variety of settings not considered in the literature.

Figures (7)

• Figure 1: A periodic graph $\mathcal{G} = \left(G_0,G_1,G_2,G_{3}\right)^{*}$, its footprint $G$, and the corresponding arena.
• Figure 2: Temporal node $(t,u)$ is a shadow corner of $(t+1, v)$.
• Figure 3: The directed acyclic graph $\mathcal{C}$ of configurations induced by $\sigma_c$ starting from $(t,x,y)$.
• Figure 4: The sets $W$ (${ \square}$) and $U$ ($\bullet$).
• Figure 5: The situation in $\mathcal{C}$ and in $\mathcal{A}$.

Theorems & Definitions (18)

• Definition 2.1: Arena
• Definition 3.1
• Lemma 3.1
• Definition 3.2
• Theorem 3.1
• Definition 3.3
• Theorem 3.2
• Theorem 3.3
• Lemma 4.1
• Lemma 4.2