Advertisement

On Dimension Partitions in Discrete Metric Spaces

  • Fabien Rebatel
  • Édouard Thiel
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7749)

Abstract

Let (W,d) be a metric space and S = {s 1s k } an ordered list of subsets of W. The distance between p ∈ W and s i  ∈ S is d(p, s i ) =  min { d(p,q) : q ∈ s i }. S is a resolving set for W if d(x, s i ) = d(y, s i ) for all s i implies x = y. A metric basis is a resolving set of minimal cardinality, named the metric dimension of (W,d). The metric dimension has been extensively studied in the literature when W is a graph and S is a subset of points (classical case) or when S is a partition of W ; the latter is known as the partition dimension problem. We have recently studied the case where W is the discrete space ℤ n for a subset of points; in this paper, we tackle the partition dimension problem for classical Minkowski distances as well as polyhedral gauges and chamfer norms in ℤ n .

Keywords

dimension partition metric dimension distance geometry discrete distance norm 

References

  1. 1.
    Buczkowski, P., Chartrand, G., Poisson, C., Zhang, P.: On k-dimensional graphs and their bases. Periodica Mathematica Hungarica 46, 9–15 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Cáceres, J., Hernando, C., Mora, M., Pelayo, I., Puertas, M.: On the metric dimension of infinite graphs. Electronic Notes in Discrete Math. 35, 15–20 (2009)CrossRefGoogle Scholar
  3. 3.
    Chappell, G.G., Gimbel, J.G., Hartman, C.: Bounds on the metric and partition dimensions of a graph. Ars Comb. 88, 349–366 (2008)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Chartrand, G., Eroh, L., Johnson, M., Oellermann, O.: Resolvability in graphs and the metric dimension of a graph. Discrete Applied Math. 105(1-3), 99–113 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Chartrand, G., Salehi, E., Zhang, P.: On the partition dimension of a graph. Congressus Numerantium 131, 55–66 (1998)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Chartrand, G., Salehi, E., Zhang, P.: The partition dimension of a graph. Aequationes Mathematicae 59, 45–54 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Harary, F., Melter, R.: On the metric dimension of a graph. Ars Combinatoria 2, 191–195 (1976)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Hardy, G., Wright, E.: An introduction to the theory of numbers, 5th edn. Oxford University Press (October 1978)Google Scholar
  9. 9.
    Hernando, C., Mora, M., Pelayo, I., Seara, C., Cáceres, J., Puertas, M.: On the metric dimension of some families of graphs. Electronic Notes in Discrete Mathematics 22, 129–133 (2005)CrossRefGoogle Scholar
  10. 10.
    Khuller, S., Raghavachari, B., Rosenfeld, A.: Landmarks in graphs. Discrete Applied Mathematics 70(3), 217–229 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Melter, R., Tomescu, I.: Metric bases in digital geometry. Computer Vision, Graphics, and Image Processing 25(1), 113–121 (1984)zbMATHCrossRefGoogle Scholar
  12. 12.
    Rebatel, F., Thiel, É.: Metric Bases for Polyhedral Gauges. In: Debled-Rennesson, I., Domenjoud, E., Kerautret, B., Even, P. (eds.) DGCI 2011. LNCS, vol. 6607, pp. 116–128. Springer, Heidelberg (2011)Google Scholar
  13. 13.
    Tomescu, I.: Discrepancies between metric dimension and partition dimension of a connected graph. Discrete Mathematics 308(22), 5026–5031 (2008)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Fabien Rebatel
    • 1
  • Édouard Thiel
    • 1
  1. 1.Laboratoire d’Informatique Fondamentale de Marseille (LIF, UMR 7279)Aix-Marseille UniversitéMarseille cedex 9France

Personalised recommendations