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.
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This research was partially supported by Abdus Salam School of Mathematical Sciences, Lahore and Higher Education Commission of Pakistan.
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Tomescu, I., Imran, M. Metric Dimension and R-Sets of Connected Graphs. Graphs and Combinatorics 27, 585–591 (2011). https://doi.org/10.1007/s00373-010-0988-8
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DOI: https://doi.org/10.1007/s00373-010-0988-8