Graphs and Combinatorics

, Volume 27, Issue 4, pp 585–591 | Cite as

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 

Mathematics Subject Classification (2000)

05C12 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

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)MATHCrossRefMathSciNetGoogle Scholar
  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)MATHCrossRefMathSciNetGoogle Scholar
  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)MATHCrossRefMathSciNetGoogle Scholar
  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)MATHMathSciNetGoogle Scholar
  5. 5.
    Garey M.R., Johnson D.S.: Computers and intractability: a guide to the theory of NP-completeness. Freeman, San Francisco (1979)MATHGoogle Scholar
  6. 6.
    Harary F., Melter R.A.: On the metric dimension of a graph. Ars Combin. 2, 191–195 (1976)MathSciNetGoogle Scholar
  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)CrossRefGoogle Scholar
  8. 8.
    Khuller S., Raghavachari B., Rosenfeld A.: Landmarks in graphs. Discrete Appl. Math. 70, 217–229 (1996)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Javaid I., Rahim M.T., Ali K.: Families of regular graphs with constant metric dimension. Utilitas Math. 75, 21–33 (2008)MATHMathSciNetGoogle Scholar
  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)MathSciNetGoogle Scholar
  11. 11.
    Slater P.J.: Dominating and reference sets in graphs. J. Math. Phys. Sci. 22, 445–455 (1988)MATHMathSciNetGoogle Scholar
  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

Copyright information

© Springer 2010

Authors and Affiliations

  1. 1.Faculty of Mathematics and Computer ScienceUniversity of BucharestBucharestRomania
  2. 2.Abdus Salam School of Mathematical SciencesGovernment College UniversityLahorePakistan

Personalised recommendations