Original Paper

Graphs and Combinatorics

, Volume 27, Issue 4, pp 585-591

First online:

Metric Dimension and R-Sets of Connected Graphs

  • Ioan TomescuAffiliated withFaculty of Mathematics and Computer Science, University of Bucharest
  • , Muhammad ImranAffiliated withAbdus Salam School of Mathematical Sciences, Government College University Email author 

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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.


Metric dimension Resolving set Diameter Clique number

Mathematics Subject Classification (2000)