Computing the Metric Dimension by Decomposing Graphs into Extended Biconnected Components

(Extended Abstract)
  • Duygu VietzEmail author
  • Stefan Hoffmann
  • Egon Wanke
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11355)


A vertex set \(U \subseteq V\) of an undirected graph \(G=(V,E)\) is a resolving set for G, if for every two distinct vertices \(u,v \in V\) there is a vertex \(w \in U\) such that the distance between u and w and the distance between v and w are different. The Metric Dimension of G is the size of a smallest resolving set for G. Deciding whether a given graph G has Metric Dimension at most k for some integer k is well-known to be NP-complete. A lot of research has been done to understand the complexity of this problem on restricted graph classes. In this paper, we decompose a graph into its so called extended biconnected components and present an efficient algorithm for computing the metric dimension for a class of graphs having a minimum resolving set with a bounded number of vertices in every extended biconnected component. Furthermore, we show that the decision problem Metric Dimension remains NP-complete when the above limitation is extended to usual biconnected components.


Graph algorithm Complexity Metric dimension Resolving set Biconnected component 


  1. 1.
    Belmonte, R., Fomin, F.V., Golovach, P.A., Ramanujan, M.: Metric dimension of bounded tree-length graphs. SIAM J. Discret. Math. 31(2), 1217–1243 (2017)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Chartrand, G., Eroh, L., Johnson, M., Oellermann, O.: Resolvability in graphs and the metric dimension of a graph. Discret. Appl. Math. 105(1–3), 99–113 (2000)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Díaz, J., Pottonen, O., Serna, M., van Leeuwen, E.J.: On the complexity of metric dimension. In: Epstein, L., Ferragina, P. (eds.) ESA 2012. LNCS, vol. 7501, pp. 419–430. Springer, Heidelberg (2012). Scholar
  4. 4.
    Epstein, L., Levin, A., Woeginger, G.J.: The (weighted) metric dimension of graphs: hard and easy cases. Algorithmica 72(4), 1130–1171 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Estrada-Moreno, A., Rodríguez-Velázquez, J.A., Yero, I.G.: The k-metric dimension of a graph. arXiv preprint arXiv:1312.6840 (2013)
  6. 6.
    Fernau, H., Heggernes, P., van’t Hof, P., Meister, D., Saei, R.: Computing the metric dimension for chain graphs. Inf. Process. Lett. 115(9), 671–676 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Foucaud, F., Mertzios, G.B., Naserasr, R., Parreau, A., Valicov, P.: Algorithms and complexity for metric dimension and location-domination on interval and permutation graphs. In: Mayr, E.W. (ed.) WG 2015. LNCS, vol. 9224, pp. 456–471. Springer, Heidelberg (2016). Scholar
  8. 8.
    Foucaud, F., Mertzios, G.B., Naserasr, R., Parreau, A., Valicov, P.: Identification, location-domination and metric dimension on interval and permutation graphs. I. Bounds. Theor. Comput. Sci. 668, 43–58 (2017)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Garey, M., Johnson, D.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman, New York (1979)zbMATHGoogle Scholar
  10. 10.
    Harary, F., Melter, R.: On the metric dimension of a graph. Ars Combinatoria 2, 191–195 (1976)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Hartung, S., Nichterlein, A.: On the parameterized and approximation hardness of metric dimension. In: 2013 IEEE Conference on Computational Complexity (CCC), pp. 266–276. IEEE (2013)Google Scholar
  12. 12.
    Hauptmann, M., Schmied, R., Viehmann, C.: Approximation complexity of metric dimension problem. J. Discret. Algorithms 14, 214–222 (2012)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hernando, C., Mora, M., Slater, P.J., Wood, D.R.: Fault-tolerant metric dimension of graphs. Convexity Discret. Struct. 5, 81–85 (2008)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Hernando, M., Mora, M., Pelayo, I., Seara, C., Cáceres, J., Puertas, M.: On the metric dimension of some families of graphs. Electron. Notes Discret. Math. 22, 129–133 (2005)CrossRefGoogle Scholar
  15. 15.
    Hoffmann, S., Wanke, E.: Metric Dimension for Gabriel unit disk graphs is NP-complete. In: Bar-Noy, A., Halldórsson, M.M. (eds.) ALGOSENSORS 2012. LNCS, vol. 7718, pp. 90–92. Springer, Heidelberg (2013). Scholar
  16. 16.
    Hoffmann, S., Elterman, A., Wanke, E.: A linear time algorithm for metric dimension of cactus block graphs. Theor. Comput. Sci. 630, 43–62 (2016)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Iswadi, H., Baskoro, E., Salman, A., Simanjuntak, R.: The metric dimension of amalgamation of cycles. Far East J. Math. Sci. (FJMS) 41(1), 19–31 (2010)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Khuller, S., Raghavachari, B., Rosenfeld, A.: Landmarks in graphs. Discret. Appl. Math. 70, 217–229 (1996)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Oellermann, O.R., Peters-Fransen, J.: The strong metric dimension of graphs and digraphs. Discret. Appl. Math. 155(3), 356–364 (2007)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Saputro, S., Baskoro, E., Salman, A., Suprijanto, D., Baca, A.: The metric dimension of regular bipartite graphs. arXiv/1101.3624 (2011).
  21. 21.
    Sebö, A., Tannier, E.: On metric generators of graphs. Math. Oper. Res. 29(2), 383–393 (2004)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Slater, P.: Leaves of trees. Congr. Numer. 14, 549–559 (1975)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Heinrich-Heine-University DuesseldorfDuesseldorfGermany

Personalised recommendations