This paper introduces a method to design observable directed multi-agent networks, that are: 1) either minimal with respect to a communications-related cost function, or 2) idem, under possible failure of direct communication between two agents. An observable multi-agent network is characterized by agents that update their states using a neighboring rule based on directed communication graph topology in order to share information about their states; furthermore, each agent can infer the initial information shared by all the agents. Sufficient conditions to ensure that 1) is satisfied are obtained by reducing the original problem to the travelling salesman problem (TSP). For the case described in 2), sufficient conditions for the existence of a minimal network are shown to be equivalent to the existence of two disjoint solutions to the TSP. The results obtained are illustrated with an example from the area of cooperative path following of multiple networked vehicles by resorting to an approximate solution to the TSP.