Abstract
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]d, and two vertices are adjacent if the Euclidean distance between them is at most r. The main result is that if r3d-1>cdlognn 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 Beveridge et al. (2012), using a different approach. Our method provides a tight O(1/r2) upper bound for the number of rounds needed to catch the robber.
Original language | English (US) |
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Pages (from-to) | 149-152 |
Number of pages | 4 |
Journal | Discrete Applied Mathematics |
Volume | 178 |
DOIs | |
State | Published - Dec 11 2014 |
Externally published | Yes |
All Science Journal Classification (ASJC) codes
- Discrete Mathematics and Combinatorics
- Applied Mathematics
Keywords
- Cops and Robbers
- Random graphs
- Vertex-pursuit games