Chasing robbers on random geometric graphs - An alternative approach

We study the vertex pursuit game of Cops and Robbers, in which cops try to capture a robber on the vertices of the graph. The minimum number of cops required to win on a given graph G is called the cop number of G. We focus on Gd(n, r), a random geometric graph in which n vertices are chosen uniformly at random and independently from [0, 1], and two vertices are adjacent if the Euclidean distance between them is at most r. The main result is that if r3d−1 > cd logn n then the cop number is 1 with probability that tends to 1 as n tends to infinity. The case d = 2 was proved earlier and independently in [4], using a different approach. Our method provides a tight O(1/r) upper bound for the number of rounds needed to catch the