Strong resolving graphs: The realization and the characterization problems

The strong resolving graph GSR of a connected graph G was introduced in Oellermann and Peters-Fransen (2007) as a tool to study the strong metric dimension of G. Basically, it was shown that the problem of finding the strong metric dimension of G can be transformed to the problem of finding the vertex cover number of GSR. Since then, several articles on the strong metric dimension of graphs which are using this tool have been published. However, the tool itself has remained unnoticed as a properly structure. In this paper, we survey the state of knowledge on the strong resolving graphs, and also derive some new results regarding its properties.