A model of mobile robots in networks with resolvability properties

A model for mobility of robots keeping the property of uniquely recognizing the vertices of a given network is considered in this work. This is made in order to detect failures or intruders, by means of dynamic vectors of distances to the set of mobile robots. We consider the smallest set of robots that can be placed in a set of nodes of a network that forms a resolving set, which is a structure of a graph such that it uniquely recognizes all the vertices of the graph by using distances. We are then focused on allowing such robots to move from one vertex to another adjacent one, through the edges of the whole graph. At each performed movement we require that the new set of covered nodes forms a resolving set. This process allows the robots to recognize all the vertices of the graph, independently on the position in which they are located. In this sense, the notion of mobile metric dimension is introduced in this article, and the study of its primary combinatorial properties is initiated. We relate this parameter with the classical metric dimension and the resolving number of graphs and compute its value for several graph classes.