On Metric Dimension of Subdivided Honeycomb Network and Aztec Diamond Network

Abstract

This paper investigates the metric dimensions of the polygonal networks, particularly, the subdivided honeycomb network, Aztec diamond as well as the subdivided Aztec diamond network. A polygon is any two-dimensional shape formed by straight lines. Triangles, quadrilaterals, pentagons, and hexagons are all representations of polygons. For instance, hexagons help us in many models to construct honeycomb network, where n is the number of hexagons from a central point to the borderline of the network. A subdivided honeycomb network $\left(\text{SHCN}\left(n\right)\right)$ is obtained by adding additional vertices on each edge of $\text{HCN}\left(n\right)$ . An Aztec diamond network $\left(\text{AZN}\left(n\right)\right)$ of order $n$ is a lattice comprises of unit squares with center $\left(a,b\right)$ satisfying $\left|a\right|+\left|b\right|\le n$ . The subdivided Aztec diamond network $\left(\text{SAZN}\left(n\right)\right)$ is obtained by adding additional vertices to each edge of $\text{AZN}\left(n\right)$ . In this work, our main aim is to establish the results to show that the metric dimensions of $\text{SHCN}\left(n\right)$ and $\text{AZN}\left(n\right)$ are 2 and 3 for $n=1$ and $n\ge 2$ , respectively. In the end, some open problems are listed with regard to metric dimensions for $k$ -subdivisions of $\text{HCN}\left(n\right)$ and $\text{AZN}\left(n\right)$ .