We consider a version of the well known graph searching problem, where a team of robots aims at clearing the contaminated edges of a graph.
We study graph searching in the model of mobile computing where
autonomous deterministic robots move between the graph nodes when
operating in asynchronous cycles of Look-Compute-Move. Moreover,
motivated by physical constraints, following some recent works, we
consider the exclusivity property, stating that no two or more robots
can occupy a same node at the same time. In addition, we assume that all
the graph elements and the robots are anonymous. Robots are oblivious
and have no sense of direction.


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.