The comparative study of resolving parameters for a family of ladder networks

Abstract

For a simple connected graph $ G = (V, E) $, a vertex $ x\in V $ distinguishes two elements (vertices or edges) $ x_1\in V, y_1 \in E $ if $ d(x, x_1)\neq d(x, y_1). $ A subset $ Q_m\subset V $ is a mixed metric generator for $ G, $ if every two distinct elements (vertices or edges) of $ G $ are distinguished by some vertex of $ Q_m. $ The minimum cardinality of a mixed metric generator for $ G $ is called the mixed metric dimension and denoted by $ dim_m(G). $ In this paper, we investigate the mixed metric dimension for different families of ladder networks. Among these families, we consider Möbius ladder, hexagonal Möbius ladder, triangular Möbius ladder network and conclude that all these families have constant-metric, edge metric and mixed metric dimension.

Keywords

• möbius ladder network
• hexagonal möbius ladder network
• triangular möbius ladder network
• mixed metric dimension
• mixed metric generator