The Simultaneous Strong Metric Dimension of Graph Families

Let ${\cal G}$ be a family of graphs defined on a common (labeled) vertex set $V$. A set $S\subset V$ is said to be a simultaneous strong metric generator for ${\cal G}$ if it is a strong metric generator for every graph of the family. The minimum cardinality among all simultaneous strong metric generators for ${\cal G}$, denoted by $Sd_s({\cal G})$, is called the simultaneous strong metric dimension of ${\cal G}$. We obtain general results on $Sd_s({\cal G})$ 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.