Metric Dimension and RSets of Connected Graphs
 Ioan Tomescu,
 Muhammad Imran
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Abstract
The Rset 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 Rsets of connected graphs. It is shown that for a connected graph G of order n and diameter 2 the number of Rsets 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 Rsets 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.
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 Title
 Metric Dimension and RSets of Connected Graphs
 Journal

Graphs and Combinatorics
Volume 27, Issue 4 , pp 585591
 Cover Date
 20110701
 DOI
 10.1007/s0037301009888
 Print ISSN
 09110119
 Online ISSN
 14355914
 Publisher
 Springer Japan
 Additional Links
 Topics
 Keywords

 Metric dimension
 Resolving set
 Diameter
 Clique number
 05C12
 Industry Sectors
 Authors

 Ioan Tomescu ^{(1)}
 Muhammad Imran ^{(2)}
 Author Affiliations

 1. Faculty of Mathematics and Computer Science, University of Bucharest, Str. Academiei, 14, 010014, Bucharest, Romania
 2. Abdus Salam School of Mathematical Sciences, Government College University, 68B, New Muslim Town, Lahore, Pakistan