Cops and Robbers on Geometric Graphs
Andrew Beveridge, Andrzej Dudek, Alan Frieze, Tobias M\"uller

TL;DR
This paper investigates the cop number in geometric graphs, establishing upper bounds for all such graphs in the plane, providing exact values for some, and analyzing probabilistic bounds for random geometric graphs.
Contribution
It proves a universal upper bound of 9 for the cop number in any connected geometric graph in the plane and refines bounds for random geometric graphs based on density.
Findings
c(G) 9 for any connected geometric graph in
c(G) 2 with high probability when r (rac{\, log n}{n})^{1/4}
c(G) 1 with high probability when r (rac{\, log n}{n})^{1/5}
Abstract
Cops and robbers is a turn-based pursuit game played on a graph . 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 denotes the minimum number of cops required to catch the robber in finite time. We study the cop number of geometric graphs. For points , and , the vertex set of the geometric graph is the graph on these points, with adjacent when . We prove that for any connected geometric graph in and we give an example of a connected geometric graph with . We improve on our upper bound for random geometric graphs that are sufficiently dense. Let denote the probability space of geometric graphs with vertices chosen uniformly and independently from…
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