# Locating a robber with multiple probes

## TL;DR

This paper investigates a pursuit-evasion game on graphs where a cop uses multiple probes to locate a moving robber, analyzing how the number of probes affects winning strategies on subdivided and general graphs.

## Contribution

It extends previous work by studying the impact of multiple probes on the cop's winning strategy, providing bounds relating probe count and graph subdivision length.

## Key findings

- Linear bound on probe number for victory on original graph
- No subexponential bound in the reverse direction
- Bound on probe number for winning on graphs with bounded degree

## Abstract

We consider a game in which a cop searches for a moving robber on a connected graph using distance probes, which is a slight variation on one introduced by Seager. Carragher, Choi, Delcourt, Erickson and West showed that for any $n$-vertex graph $G$ there is a winning strategy for the cop on the graph $G^{1/m}$ obtained by replacing each edge of $G$ by a path of length $m$, if $m\geq n$. The present authors showed that, for all but a few small values of $n$, this bound may be improved to $m\geq n/2$, which is best possible. In this paper we consider the natural extension in which the cop probes a set of $k$ vertices, rather than a single vertex, at each turn. We consider the relationship between the value of $k$ required to ensure victory on the original graph and the length of subdivisions required to ensure victory with $k=1$. We give an asymptotically best-possible linear bound in one direction, but show that in the other direction no subexponential bound holds. We also give a bound on the value of $k$ for which the cop has a winning strategy on any (possibly infinite) connected graph of maximum degree $\Delta$, which is best possible up to a factor of $(1-o(1))$.