On Metric Dimension and Fault Tolerant Metric Dimension of Some Chemical Structures

In chemistry, customary depiction for the structure of chemical compounds are graphs where the vertices and edges represents the atoms and bonds types, respectively. Resolving set aleph is an ordered subset of vertices of a graph G, in which each vertex of G is distinctively determined by its distance to the vertices in aleph. The cardinality of a minimum resolving set is called the metric dimension of G. A resolving set aleph f of G is fault-tolerant if aleph f set minus v is also a resolving set, for every v in aleph f. Resolving set allows to obtain a unique representation for chemical structures. In particular, they were used in pharmaceutical research for discovering patterns common to a variety of drugs. In this paper we determine the exact metric and fault tolerant metric dimension of cyclic hexagonal square chain, linear phenylene, and linear heptagons structure.