#### Date of Original Version

8-2-2011

#### Type

Article

#### Rights Management

© Cambridge University Press

#### Abstract or Description

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} *∈ ℝ

^{2}, and

*r*∈ ℝ

^{+}, the vertex set of the geometric graph

*G(x*

^{1}, . . .,

*x*) is the graph on these

_{n}; r*n*points, with

*x*adjacent when ∥

_{i}, x_{j}*x*−

_{i}*x*∥ ≤

_{j}*r*. We prove that

*c*(

*G*) ≤ 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 (

*n,r*) denote the probability space of geometric graphs with

*n*vertices chosen uniformly and independently from [0,1]

^{2}. For

*G*∈ (

*n,r*), we show that with high probability (w.h.p.), if

*r*≥

*K*

_{1}(log

*n/n*)

^{1/4}then

*c*(

*G*) ≤ 2, and if

*r*≥

*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 (

*n,r*): if

*r*≤

*K*

_{3}log

*n*/ then

*c*(

*G*) > 1 w.h.p., where

*K*

_{3}> 0 is an absolute constant.

#### DOI

10.1017/S0963548312000338

#### Published In

Combinatorics, Probability and Computing, 21, 6, 816-834.