Our objective is to characterize for a graph $G$, the set of integers $k$ such that graph searching can be achieved by a team of $k$ robots starting from any $k$ distinct nodes in $G$. Our main result consists in a full characterization for any asymmetric tree. Towards providing a characterization in the general case, including trees with non-trivial automorphisms, we provide a set of positive and negative results, including a full characterization for any line. All our positive results are based on the design of algorithms enabling perpetual graph searching to be achieved with the desired number of robots.

We prove that, in addition to the distributed nature of our setting, the exclusivity property has a significant impact on the nature of the graph searching problem. Hence, the design of the algorithms requires to invent new methods.