Simultaneous resolvability in graph families Y. Ram´ ırez-Cruz 1 , O. R. Oellermann 2 , and J. A. Rodr´ ıguez-Vel´azquez 1 1 Departament d’Enginyeria Inform` atica i Matem` atiques, Universitat Rovira i Virgili, Av. Pa¨ ısos Catalans 26, 43007 Tarragona, Spain {yunior.ramirez,juanalberto.rodriguez}@urv.cat 2 Department of Mathematics and Statistics University of Winnipeg, 515 Portage Avenue, Winnipeg, MB, Canada. o.oellermann@uwinnipeg.ca Abstract. A set S ⊆ V is said to be a simultaneous metric generator for a graph family G = {G 1 ,G 2 ,...,G k }, defined on a common vertex set, if it is a generator for every graph of the family. A minimum simultaneous metric generator is called a simultaneous metric basis, and its cardinality the simultaneous metric dimension of G. We study the properties of simultaneous metric generators and simultaneous metric bases, and calculate closed formulae or tight bounds for the simultaneous metric dimension of several graph families. Key words: simultaneous metric generator, simultaneous metric basis, simultaneous metric dimension. 1 Introduction A generator of a metric space is a set S of points in the space with the property that every point of the space is uniquely determined by its distances from the elements of S . Given a simple and connected graph G =(V,E), we consider the function d G : V × V → N, where d G (x, y) is the length of a shortest path between u and v and N is the set of non-negative integers. Clearly, (V,d G ) is a metric space. A vertex v ∈ V is said to distinguish two vertices x and y if d G (v,x) = d G (v,y). A set S ⊆ V is said to be a metric generator for G if any pair of vertices of G is distinguished by some element of S . A minimum metric generator is called a metric basis, and its cardinality the metric dimension of G, denoted by dim(G). The concept of metric dimension of a graph was introduced by Slater in [8], where metric generators were called locating sets, and, independently, by Harary and Melter in [3], where metric generators were called resolving sets. Applications of the metric dimension to the navigation of robots in networks are discussed in [7] and applications to chemistry in [5,6]. Now we present the navigation problem proposed in [7] where navigation was studied in a graph-structured framework in which the navigating agent