Abstract
The R-set relative to a pair of distinct vertices of a connected gr aph G is the set of vertices whose distances to these vertices are distinct. In this paper R-sets are used to show that metric dimension dim(Nen) = 3 when n is odd and 2 otherwise, where Nen is the necklace graph of order 2n+ 2. It is also shown that the exchange property of the bases in a vector space does not hold for minimal resolving sets of Nen if n is even.
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