# Metric Dimension and *R*-Sets of Connected Graphs

- First Online:

- Received:
- Revised:

DOI: 10.1007/s00373-010-0988-8

- Cite this article as:
- Tomescu, I. & Imran, M. Graphs and Combinatorics (2011) 27: 585. doi:10.1007/s00373-010-0988-8

## Abstract

The *R*-set relative to a pair of distinct vertices of a connected graph *G* is the set of vertices whose distances to these vertices are distinct. This paper deduces some properties of *R*-sets of connected graphs. It is shown that for a connected graph *G* of order *n* and diameter 2 the number of *R*-sets equal to *V*(*G*) is bounded above by \({\lfloor n^{2}/4\rfloor}\) . It is conjectured that this bound holds for every connected graph of order *n*. A lower bound for the metric dimension *dim*(*G*) of *G* is proposed in terms of a family of *R*-sets of *G* having the property that every subfamily containing at least *r* ≥ 2 members has an empty intersection. Three sufficient conditions, which guarantee that a family \({\mathcal{F}=(G_{n})_{n\geq 1}}\) of graphs with unbounded order has unbounded metric dimension, are also proposed.