Computation of metric dimension and partition dimension of nanotubes

The partition dimension denoted by pd and metric dimension denoted by dim of a connected graph are related as pd ≤ dim +1. However, the partition dimension may be much smaller than the metric dimension and this phenomena is called a discrepancy between metric dimension and partition dimension. In this paper, we study the metric dimension (location number) and partition dimension of some infinite nanotubes generated by tiling of the plane. We prove that these nanotubes have discrepancy between their metric dimension and partition dimension. It is natural to ask for the characterization of graphs having discrepancies between their metric dimension and partition dimension. It is also shown that there exist induced subgraphs of these nanostructures having metric dimension n as well as having constant metric dimension.