Graphs and Combinatorics

, Volume 27, Issue 4, pp 585–591

# Metric Dimension and R-Sets of Connected Graphs

Original Paper

## 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.

### Keywords

Metric dimension Resolving set Diameter Clique number

05C12

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### References

1. 1.
Buczkowski P.S., Chartrand G., Poisson C., Zhang P.: On k-dimensional graphs and their bases. Periodica Math. Hung. 46(1), 9–15 (2003)
2. 2.
Cáceres J., Hernando C., Mora M., Pelayo I.M., Puertas M.L., Seara C., Wood D.R.: On the metric dimension of Cartesian products of graphs. SIAM J. Discrete Math. 21, 423–441 (2007)
3. 3.
Chartrand G., Eroh L., Johnson M.A., Oellermann O.R.: Resolvability in graphs and metric dimension of a graph. Discrete Appl. Math. 105, 99–113 (2000)
4. 4.
Currie J.D., Oellermann O.R.: The metric dimension and metric independence of a graph. J. Combin. Math. Combin. Comput. 39, 157–167 (2001)
5. 5.
Garey M.R., Johnson D.S.: Computers and intractability: a guide to the theory of NP-completeness. Freeman, San Francisco (1979)
6. 6.
Harary F., Melter R.A.: On the metric dimension of a graph. Ars Combin. 2, 191–195 (1976)
7. 7.
Hernando C., Mora M., Pelayo I.M., Seara C., Wood D.R.: Extremal graph theory for metric dimension and diameter. Electron. Notes Discrete Math. 29, 339–343 (2009)
8. 8.
Khuller S., Raghavachari B., Rosenfeld A.: Landmarks in graphs. Discrete Appl. Math. 70, 217–229 (1996)
9. 9.
Javaid I., Rahim M.T., Ali K.: Families of regular graphs with constant metric dimension. Utilitas Math. 75, 21–33 (2008)
10. 10.
Slater P.J.: Leaves of trees, proceedings of 6th southeastern conference on combinatorics, graph theory and computing. Congr. Numer. 14, 549–559 (1975)
11. 11.
Slater P.J.: Dominating and reference sets in graphs. J. Math. Phys. Sci. 22, 445–455 (1988)
12. 12.
Tomescu, I., Imran, M.: On metric and partition dimensions of some infinite regular graphs. Bull. Math. Soc. Sci. Math. Roumanie 52(100), 4, 461–472 (2009)Google Scholar