Computation of Edge Resolvability of Benzenoid Tripod Structure

Abstract

In chemistry, graphs are commonly used to show the structure of chemical compounds, with nodes and edges representing the atom and bond types, respectively. Edge resolving set ${\lambda }_e$ is an ordered subset of nodes of a graph $C$ , in which each edge of $C$ is distinctively determined by its distance vector to the nodes in $\lambda$ . The cardinality of a minimum edge resolving set is called the edge metric dimension of $C$ . An edge resolving set $L_{e,f}$ of $C$ is fault-tolerant if ${\lambda }_{e,f}\mathrm{\setminus }b$ is also an edge resolving set, for every $b$ in ${\lambda }_{e,f}$ . Resolving set allows obtaining 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 edge metric and fault-tolerant edge metric dimension of benzenoid tripod structure and proved that both parameters are constant.