The Strong Resolving Graph and the Strong Metric Dimension of Cactus Graphs

Abstract

A vertexwof a connected graphGstrongly resolves two distinct verticesu,v is an element of V(G), if there is a shortestu,wpath containingv, or a shortestv,wpath containingu. A setSof vertices ofGis astrong resolving setforGif every two distinct vertices ofGare strongly resolved by a vertex ofS. The smallest cardinality of a strong resolving set forGis called thestrong metric dimensionofG. To study the strong metric dimension of graphs, a very important role is played by a structure of graphs called the strong resolving graph In this work, we obtain the strong metric dimension of some families of cactus graphs, and along the way, we give several structural properties of the strong resolving graphs of the studied families of cactus graphs.