Metric Dimension of Cayley Graphs Symmetric Groups and Their Transpositions

Pretend that you cannot remember where you parked your car in the parking lot of the grocery store, but you do remember some of the cars parked near you. One could construct a graph based on your memory of the cars and then use the idea of the metric dimension to find your car. The metric dimension was introduced by PJ Slater in 1975 and has since been applied in fields such as chemistry, optimization, navigation, and more. There is no general/standard metric dimension for every graph, however, there are known metric dimensions for families of graphs. In this paper we study the metric dimension of Cayley graphs, which are graphs based on groups that have convenient algebraic properties. Our main goal is to find the metric dimension of the Cayley graph associated with the symmetric group $S_4$ and its set of transpositions $T_4$.