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Cops and Robbers on Geometric Graphs

Published online by Cambridge University Press:  16 August 2012

ANDREW BEVERIDGE ,
ANDRZEJ DUDEK ,
ALAN FRIEZE  and
TOBIAS MÜLLER
ANDREW BEVERIDGE
Affiliation:
Department of Mathematics, Statistics and Computer Science, Macalester College, Saint Paul, MN 55015, USA (e-mail: abeverid@macalester.edu)
ANDRZEJ DUDEK
Affiliation:
Department of Mathematics, Western Michigan University, Kalamazoo, MI 49008, USA (e-mail: andrzej.dudek@wmich.edu)
ALAN FRIEZE
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA (e-mail: alan@random.math.cmu.edu)
TOBIAS MÜLLER
Affiliation:
Centrum voor Wiskunde en Informatica, 1098 XG Amsterdam, the Netherlands (e-mail: tobias@cwi.nl)
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Abstract

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 x1, . . ., xn ∈ ℝ2, and r ∈ ℝ+, the vertex set of the geometric graph G(x1, . . ., xn; r) is the graph on these n points, with xi, xj adjacent when ∥xi − xj∥ ≤ r. We prove that c(G) ≤ 9 for any connected geometric graph G in ℝ2 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 ≥ K1 (log n/n)1/4 then c(G) ≤ 2, and if r ≥ K2(log n/n)1/5 then c(G) = 1, where K1, K2 > 0 are absolute constants. Finally, we provide a lower bound near the connectivity regime of (n,r): if r ≤ K3 log n/ then c(G) > 1 w.h.p., where K3 > 0 is an absolute constant.

Keywords

Primary 05C57Secondary 60D0505C80
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 21 , Issue 6 , November 2012 , pp. 816 - 834
Copyright
Copyright © Cambridge University Press 2012

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