Advertisement

International Workshop on Combinatorial Algorithms

IWOCA 2014: Combinatorial Algorithms pp 330-337 | Cite as

Metric Dimension for Amalgamations of Graphs

  • Rinovia SimanjuntakEmail author
  • Saladin Uttunggadewa
  • Suhadi Wido Saputro
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8986)

Abstract

A set of vertices S resolves a graph G if every vertex is uniquely determined by its vector of distances to the vertices in S. The metric dimension of G is the minimum cardinality of a resolving set of G.

Let \(\{G_1, G_2, \ldots , G_n\}\) be a finite collection of graphs and each \(G_i\) has a fixed vertex \(v_{0_i}\) or a fixed edge \(e_{0_i}\) called a terminal vertex or edge, respectively. The vertex-amalgamation of \(G_1, G_2, \ldots , G_n\), denoted by \(Vertex-Amal\{G_i;v_{0_i}\}\), is formed by taking all the \(G_i\)’s and identifying their terminal vertices. Similarly, the edge-amalgamation of \(G_1, G_2, \ldots , G_n\), denoted by \(Edge-Amal\{G_i;e_{0_i}\}\), is formed by taking all the \(G_i\)’s and identifying their terminal edges.

Here we study the metric dimensions of vertex-amalgamation and edge-amalgamation for finite collection of arbitrary graphs. We give lower and upper bounds for the dimensions, show that the bounds are tight, and construct infinitely many graphs for each possible value between the bounds.

References

  1. 1.
    Bailey, R.F., Cameron, P.J.: Base size, metric dimension and other invariants of groups and graphs. Bull. Lond. Math. Soc. 43, 209–242 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Bollobas, B., Mitsche, D., Pralat, P.: Metric dimension for random graphs. Electron. J. Comb. 20, \(\sharp \)P1 (2013)Google Scholar
  3. 3.
    Buczkowski, P.S., Chartrand, G., Poisson, C., Zhang, P.: On \(k\)-dimensional graphs and their bases. Period. Math. Hung. 46, 9–15 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Caceres, 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)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Chartrand, G., Eroh, L., Johnson, M.A., Oellermann, O.R.: Resolvability in graphs and the metric dimension of a graph. Discrete Appl. Math. 105, 99–113 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    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) CrossRefGoogle Scholar
  7. 7.
    Erdős, P., Rényi, A.: On two problems of information theory. Magyar Tud. Akad. Mat. Kutat Int. Kzl. 8, 229–243 (1963)Google Scholar
  8. 8.
    Feng, M., Wang, K.: On the metric dimension and fractional metric dimension of the hierarchical product of graphs. Appl. Anal. Discrete Math. 7, 302–313 (2013)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Feng, M., Xu, M., Wang, K.: On the metric dimension of line graphs. Discrete Appl. Math. 161, 802–805 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Garey, M.R., Johnson, D.S.: Computers and Intractibility: A Guide to the Theory of NP Completeness. W.H. Freeman and Company, San Francisco (1979) zbMATHGoogle Scholar
  11. 11.
    Harary, F., Melter, R.A.: On the metric dimension of a graph. Ars Comb. 2, 191–195 (1976)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Hernando, C., Mora, M., Pelayo, I.M., Seara, C., Wood, D.R.: Extremal graph theory for metric dimension and diameter. Electron. J. Comb. 17 \(\sharp \)R30 (2010)Google Scholar
  13. 13.
    Iswadi, H., Baskoro, E.T., Salman, A.N.M., Simanjuntak, R.: The metric dimension of amalgamation of cycles. Far East J. Math. Sci. 41, 19–31 (2010)zbMATHMathSciNetGoogle Scholar
  14. 14.
    Iswadi, H., Baskoro, E.T., Simanjuntak, R.: On the metric dimension of corona product of graphs. Far East J. Math. Sci. 52, 155–170 (2011)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Jannesari, M., Omoomi, B.: The metric dimension of the lexicographic product of graphs. Discrete Math. 312, 3349–3356 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Jannesari, M., Omoomi, B.: Characterization of \(n\)-vertex graphs with metric dimension \(n-3\). Math. Bohemica 139, 1–23 (2014)zbMATHMathSciNetGoogle Scholar
  17. 17.
    Klein, D.J., Yi, E.: A comparison on metric dimension of graphs, line graphs, and line graphs of the subdivision graphs. European J. Pure Appl. Math. 5, 302–316 (2012)MathSciNetGoogle Scholar
  18. 18.
    Khuller, S., Raghavachari, B., Rosenfeld, A.: Landmarks in graphs. Discrete Appl. Math. 70, 217–229 (1996)zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Manuel, P.D., Abd-El-Barr, M.I., Rajasingh, I., Rajan, B.: An efficient representation of Benes networks and its applications. J. Discrete Algorithms 6, 11–19 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Poisson, C., Zhang, P.: The metric dimension of unicyclic graphs. J. Comb. Math. Comb. Comput. 40, 17–32 (2002)zbMATHMathSciNetGoogle Scholar
  21. 21.
    Saputro, S.W., Simanjuntak, R., Uttunggadewa, S., Assiyatun, H., Baskoro, E.T., Salman, A.N.M., Baća, M.: The metric dimension of the lexicographic product of graphs. Discrete Math. 313, 1045–1051 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Simanjuntak, R., Assiyatun, H., Baskoroputro, H., Iswadi, H., Setiawan, Y., Uttunggadewa, S.: Graphs with relatively constant metric dimensions (preprint)Google Scholar
  23. 23.
    Simanjuntak, R., Murdiansyah, D.: Metric dimension of amalgamation of some regular graphs (preprint)Google Scholar
  24. 24.
    Slater, P.J.: Leaves of trees. Congr. Numer. 14, 549–559 (1975)MathSciNetGoogle Scholar
  25. 25.
    Tavakoli, M., Rahbarnia, F., Ashrafi, A.R.: Distribution of some graph invariants over hierarchical product of graphs. App. Math. Comput. 220, 405–413 (2013)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Yero, I.G., Kuziak, D., Rodriguez-Velázquez, J.A.: On the metric dimension of corona product graphs. Comput. Math. Appl. 61, 2793–2798 (2011)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Rinovia Simanjuntak
    • 1
    Email author
  • Saladin Uttunggadewa
    • 1
  • Suhadi Wido Saputro
    • 1
  1. 1.Combinatorial Mathematics Research Group, Faculty of Mathematics and Natural SciencesInstitut Teknologi BandungBandungIndonesia

Personalised recommendations