Metric Bases for Polyhedral Gauges

  • Fabien Rebatel
  • Édouard Thiel
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6607)


Let (W,d) be a metric space. A subset S ⊆ W is a resolving set for W if d(x,p) = d(y,p) for all p ∈ S implies x = y. A metric basis is a resolving set of minimal cardinality, named the metric dimension (of W). Metric bases and dimensions have been extensively studied for graphs with the intrinsic distance, as well as in the digital plane with the city-block and chessboard distances. We investigate these concepts for polyhedral gauges, which generalize in the Euclidean space the chamfer norms in the digital space.


metric basis metric dimension resolving set polyhedral gauge chamfer norms discrete distance distance geometry 


  1. 1.
    Berger, M.: Géométrie. Nathan (1990)Google Scholar
  2. 2.
    Blumenthal, L.M.: Theory and applications of distance geometry. Chelsea publishing company, New York (1970)zbMATHGoogle Scholar
  3. 3.
    Buczkowski, P.S., Chartrand, G., Poisson, C., Zhang, P.: On k-dimensional graphs and their bases. Periodica Mathematica Hungarica 46, 9–15 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cáceres, J., Hernando, C., Mora, M., Pelayo, I.M., Puertas, M.L.: On the metric dimension of infinite graphs. Electronic Notes in Discrete Math. 35, 15–20 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chartrand, G., Eroh, L., Johnson, M.A., Oellermann, O.R.: Resolvability in graphs and the metric dimension of a graph. Discrete Applied Math. 105(1-3), 99–113 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chartrand, G., Salehi, E., Zhang, P.: On the partition dimension of a graph. Congressus Numerantium 131, 55–66 (1998)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Chartrand, G., Salehi, E., Zhang, P.: The partition dimension of a graph. Aequationes Mathematicae 59, 45–54 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chartrand, G., Zhang, P.: The forcing dimension of a graph. Mathematica Bohemica 126(4), 711–720 (2001)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Harary, F., Melter, R.A.: On the metric dimension of a graph. Ars Combinatoria 2, 191–195 (1976)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Hardy, G.H., Wright, E.M.: An introduction to the theory of numbers, 5th edn. Oxford Science Pub. (1979)Google Scholar
  11. 11.
    Hernando, C., Mora, M., Pelayo, I.M., Seara, C., Cáceres, J., Puertas, M.L.: On the metric dimension of some families of graphs. Electronic Notes in Discrete Mathematics 22, 129–133 (2005)CrossRefzbMATHGoogle Scholar
  12. 12.
    Hulin, J.: Axe médian discret: propriétés arithmétiques et algorithmes. Thèse de Doctorat, Aix-Marseille Université (November 2009),
  13. 13.
    Khuller, S., Raghavachari, B., Rosenfeld, A.: Landmarks in graphs. Discrete Applied Mathematics 70(3), 217–229 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Melter, R.A., Tomescu, I.: Metric bases in digital geometry. Computer Vision, Graphics, and Image Processing 25(1), 113–121 (1984)CrossRefzbMATHGoogle Scholar
  15. 15.
    Normand, N., Evenou, P.: Medial axis lookup table and test neighborhood computation for 3D chamfer norms. Pattern Recognition 42(10), 2288–2296 (2009)CrossRefzbMATHGoogle Scholar
  16. 16.
    Slater, P.J.: Leaves of trees. Congressus Numerantium 14, 549–559 (1975)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Thiel, E.: Géométrie des distances de chanfrein. Habilitation à Diriger des Recherches, Université de la Méditerranée, Aix-Marseille 2 (December 2001),
  18. 18.
    Tomescu, I.: Discrepancies between metric dimension and partition dimension of a connected graph. Discrete Mathematics 308(22), 5026–5031 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Wikipedia: Gallery of named graphs,

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Fabien Rebatel
    • 1
  • Édouard Thiel
    • 1
  1. 1.Laboratoire d’Informatique Fondamentale de Marseille (LIF, UMR 6166)Aix-Marseille UniversitéMarseille cedex 9France

Personalised recommendations