Summary: In this paper we investigate the basic problem of Exploration of a graph by a group of identical mobile computational entities, called robots, operating autonomously and asynchronously. In particular we are concerned with what graphs can be explored, and how, if the robots do not remember the past and have no explicit means of communication. This model of robots is used when the spatial universe in which the robots operate is continuous (e.g., a curve, a polygonal region, a plane, etc.). The case when the spatial universe is discrete (i.e., a graph) has been also studied but only for the classes of acyclic graphs and of simple cycles. In this paper we consider networks of arbitrary topology modeled as connected graphs with local orientation (locally distinct edge labels). We concentrate on class \({\mathcal H}_k\) of asymmetric configurations with \(k\) robots. Our results indicate that the explorability of graphs in this class depends on the number \(k\) of robots participating in the exploration. In particular, exploration is impossible for \(k<3\) robots. When there are only \(k=3\) robots, only a subset of \({\mathcal H}_3\) can be explored; we provide a complete characterization of the networks that can be explored. When there are \(k=4\) robots, we prove that all networks in \({\mathcal H}_4\) can be explored. Finally, we prove that for any odd \(k>4\) all networks in \({\mathcal H}_k\) can be explored by presenting a general algorithm. The determination of which networks can be explored when \(k>4\) is even, is still open but can be reduced to the existence of a gathering algorithm for \({\mathcal H}_k\).
For the entire collection see [Zbl 1200.68024].