The Simultaneous Strong Metric Dimension of Graph Families

Let G be a family of graphs defined on a common (labelled) vertex set V. A set S subset of V is said to be a simultaneous strong metric generator for G if it is a strong metric generator for every graph of the family. The minimum cardinality among all simultaneous strong metric generators for G, denoted by Sd(s) (G), is called the simultaneous strong metric dimension of G. We obtain general results on Sd(s) (G) for arbitrary families of graphs, with special emphasis on the case of families composed by a graph and its complement. In particular, it is shown that the problem of finding the simultaneous strong metric dimension of families of graphs is NP-hard, even when restricted to families of trees.