Cops and Robbers on Geometric Graphs

Cops and robbers is a turn-based pursuit game played on a graph $G$. One robber is pursued by a set of cops. In each round, these agents move between vertices along the edges of the graph. The cop number $c(G)$ denotes the minimum number of cops required to catch the robber in finite time. We study the cop number of geometric graphs. For points $x_1, ..., x_n \in \R^2$, and $r \in \R^+$, the vertex set of the geometric graph $G(x_1, ..., x_n; r)$ is the graph on these $n$ points, with $x_i, x_j$ adjacent when $\norm{x_i -x_j} \leq r$. We prove that $c(G) \leq 9$ for any connected geometric graph $G$ in $R^2$ and we give an example of a connected geometric graph with $c(G) = 3$. We improve on our upper bound for random geometric graphs that are sufficiently dense. Let $G(n,r)$ denote the probability space of geometric graphs with $n$ vertices chosen uniformly and independently from $[0,1]^2$. For $G \in G(n,r)$, we show that with high probability (whp), if $r \geq K_1 (\log n/n)^{1/4}$, then $c(G) \leq 2$, and if $r \geq K_2(\log n/n)^{1/5}$, then $c(G) = 1$ where $K_1, K_2 > 0$ are absolute constants. Finally, we provide a lower bound near the connectivity regime of $G(n,r)$: if $r \leq K_3 \log n / \sqrt{n}$ then $c(G) > 1$ whp, where $K_3 > 0$ is an absolute constant.