Metric Dimension and R-Sets of Connected Graphs

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.