How to Hunt an Invisible Rabbit on a Graph

TL;DR

This paper analyzes the Hunters & Rabbit game on graphs, determining the minimum hunters needed on grids and trees, revealing bounds and exact values for different graph structures.

Contribution

It provides exact and asymptotic bounds for the minimum number of hunters required to catch an invisible rabbit on grids and trees.

Findings

Minimum hunters on an (n×m)-grid is  min{n,m}/2+1.

Number of hunters on n-vertex trees is between (log n/log log n) and (log n).

The results establish bounds for hunter numbers on specific graph classes.

Abstract

We investigate Hunters & Rabbit game, where a set of hunters tries to catch an invisible rabbit that slides along the edges of a graph. We show that the minimum number of hunters required to win on an (n\times m)-grid is \lfloor min{n,m}/2\rfloor+1. We also show that the extremal value of this number on n-vertex trees is between \Omega(log n/log log n) and O(log n).