Cop numbers of periodic graphs

Abstract

A \emph{periodic graph} ${\cal G}=(G_0, G_1, G_2, \dots)$ with period $p$ is an infinite periodic sequence of graphs $G_i = G_{i + p} = (V,E_i)$, where $i \geq 0$. The graph $G=(V,\cup_i E_i)$ is called the footprint of ${\cal G}$. 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 \emph{cop number} is also the minimum number of cops sufficient for capturing the robber. We study the connections and distinctions between the cop number $c({\cal G})$ of a periodic graph ${\cal G}$ and the cop number $c(G)$ of its footprint $G$ and establish several facts. For instance, we show that the smallest periodic graph with $c({\cal G}) = 3$ 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 cop numbers of a periodic graph ${\cal G}$, the subgraphs $G_i$ and its footprint $G$ can be loosely tied. Based on these results, we derive upper bounds on the cop number of a periodic graph from properties of its footprint such as its treewidth.

Figures (11)

• Figure 1: A periodic graph $\mathcal{G} = \left(G_0,G_1,G_2\right)^{*}$ with its footprint $G$ and corresponding arena
• Figure 2: An arena with a temporal corner $(0,c)$ of $(1,a)$
• Figure 3: Periodic graph used in \ref{['prop:hypercubeQ3_3copwin']}
• Figure 4: $c(\mathcal{G}[\set{a,b,c,d}]) > c(\mathcal{G})$
• Figure 5: The $(1,1,2)$-copwin periodic graph presented in \ref{['conj:112_periodicgraph']}

Theorems & Definitions (36)

• definition \@thmcounterdefinition: Arena
• lemma \@thmcounterlemma
• proof
• proposition \@thmcounterproposition
• proof
• proposition \@thmcounterproposition: 3-copwin cube periodic graph
• proof
• theorem \@thmcountertheorem
• proposition \@thmcounterproposition
• proof