Computing vertex resolvability of benzenoid tripod structure

Abstract

<abstract><p>In this paper, we determine the exact metric and fault-tolerant metric dimension of the benzenoid tripod structure. We also computed the generalized version of this parameter and proved that all the parameters are constant. Resolving set $ {L} $ is an ordered subset of nodes of a graph $ {C} $, in which each vertex of $ {C} $ is distinctively determined by its distance vector to the nodes in $ {L} $. The cardinality of a minimum resolving set is called the metric dimension of $ {C} $. A resolving set $ L_{f} $ of $ {C} $ is fault-tolerant if $ {L}_{f}\setminus{b} $ is also a resolving set, for every $ {b} $ in $ {L}_{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. The above definitions are based on the hypothesis of chemical graph theory and it is a customary depiction of chemical compounds in form of graph structures, where the node and edge represents the atom and bond types, respectively.</p></abstract>