We establish a classification of decision problems that are to be solved by mobile agents operating in unlabeled graphs, using a deterministic protocol. The classification is with respect to the ability of a team of agents to solve the problem, possibly with the aid of additional information. In particular, our focus is on studying differences between the decidability of a decision problem by agents and its verifiability when a certificate for a positive answer is provided to the agents. We show that the class MAV of mobile agents verifiable problems is much wider than the class MAD of mobile agents decidable problems. Our main result shows that there exist natural MAV-complete problems: the most difficult problems in this class, to which all problems in MAV are reducible. Our construction of a MAV-complete problem involves two main ingredients in mobile agents computability: the topology of the quotient graph and the number of operating agents. Beyond the class MAV we show that, for a single agent, three natural oracles yield a strictly increasing chain of relative decidability classes.