Adding a referee to an interconnection network: What can(not) be computed in one round

In this paper we ask which properties of a distributed network can be computed from a few amount of local information provided by its nodes. Each of these n nodes - which only knows its own ID and the IDs of its neighbors - is allowed to send a message of O(log n) bits to some central entity, called the referee. Is it possible for the referee to decide some basic structural properties of the network topology G? We show that simple questions like "does G contain a triangle (resp., a square)?" cannot be solved in general. On the other hand, the referee can decode the messages in order to have full knowledge of G when G belongs to many graph classes such as planar graphs, bounded treewidth graphs and, more generally, bounded degeneracy graphs. We leave open questions related to the connectivity and the diameter (of arbitrary graphs).