How to hunt an invisible rabbit on a graph

We investigate Hunters & Rabbit game on graphs, where a set of hunters tries to catch an invisible rabbit that is forced to slide along an edge of a graph at every round. We show that the minimum number of hunters required to win on an (n×m)(n×m)-grid is ⌊min{n,m}2⌋+1. We also show that the extremal value of this number on nn-vertex trees is between Ω(logn/loglogn)Ω(logn/loglogn) and O(logn)O(logn).