Skip to main content
SpringerLink
Log in

Notions of Metric Dimension of Corona Products: Combinatorial and Computational Results

  • Conference paper
Computer Science - Theory and Applications (CSR 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8476))

Included in the following conference series:

Abstract

The metric dimension is quite a well-studied graph parameter. Recently, the adjacency metric dimension and the local metric dimension have been introduced. We combine these variants and introduce the local adjacency metric dimension. We show that the (local) metric dimension of the corona product of a graph of order n and some non-trivial graph H equals n times the (local) adjacency metric dimension of H. This strong relation also enables us to infer computational hardness results for computing the (local) metric dimension, based on according hardness results for (local) adjacency metric dimension that we also give.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bailey, R.F., Meagher, K.: On the metric dimension of Grassmann graphs. Discrete Mathematics & Theoretical Computer Science 13, 97–104 (2011)

    MATH  MathSciNet  Google Scholar 

  2. Brigham, R.C., Chartrand, G., Dutton, R.D., Zhang, P.: Resolving domination in graphs. Mathematica Bohemica 128(1), 25–36 (2003)

    MATH  MathSciNet  Google Scholar 

  3. Buczkowski, P.S., Chartrand, G., Poisson, C., Zhang, P.: On k-dimensional graphs and their bases. Periodica Mathematica Hungarica 46(1), 9–15 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  4. Calabro, C., Impagliazzo, R., Paturi, R.: The complexity of satisfiability of small depth circuits. In: Chen, J., Fomin, F.V. (eds.) IWPEC 2009. LNCS, vol. 5917, pp. 75–85. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  5. Charon, I., Hudry, O., Lobstein, A.: Minimizing the size of an identifying or locating-dominating code in a graph is NP-hard. Theoretical Computer Science 290(3), 2109–2120 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  6. Chartrand, G., Saenpholphat, V., Zhang, P.: The independent resolving number of a graph. Mathematica Bohemica 128(4), 379–393 (2003)

    MATH  MathSciNet  Google Scholar 

  7. Colbourn, C.J., Slater, P.J., Stewart, L.K.: Locating dominating sets in series parallel networks. Congressus Numerantium 56, 135–162 (1987)

    MathSciNet  Google Scholar 

  8. Crowston, R., Gutin, G., Jones, M., Saurabh, S., Yeo, A.: Parameterized study of the test cover problem. In: Rovan, B., Sassone, V., Widmayer, P. (eds.) MFCS 2012. LNCS, vol. 7464, pp. 283–295. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  9. Díaz, J., Pottonen, O., Serna, M.J., 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)

    Chapter  Google Scholar 

  10. Feng, M., Wang, K.: On the metric dimension of bilinear forms graphs. Discrete Mathematics 312(6), 1266–1268 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  11. Frucht, R., Harary, F.: On the corona of two graphs. Aequationes Mathematicae 4, 322–325 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  12. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York (1979)

    MATH  Google Scholar 

  13. Guo, J., Wang, K., Li, F.: Metric dimension of some distance-regular graphs. Journal of Combinatorial Optimization, 1–8 (2012)

    Google Scholar 

  14. Gutin, G., Muciaccia, G., Yeo, A.: (non-)existence of polynomial kernels for the test cover problem. Information Processing Letters 113(4), 123–126 (2013)

    Google Scholar 

  15. Hammack, R., Imrich, W., Klavžar, S.: Handbook of product graphs. Discrete Mathematics and its Applications, 2nd edn. CRC Press (2011)

    Google Scholar 

  16. Harary, F., Melter, R.A.: On the metric dimension of a graph. Ars Combinatoria 2, 191–195 (1976)

    MATH  MathSciNet  Google Scholar 

  17. Hartung, S., Nichterlein, A.: On the parameterized and approximation hardness of metric dimension. In: Proceedings of the 28th IEEE Conference on Computational Complexity (CCC 2013), pp. 266–276. IEEE (2013)

    Google Scholar 

  18. Haynes, T.W., Henning, M.A., Howard, J.: Locating and total dominating sets in trees. Discrete Applied Mathematics 154(8), 1293–1300 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  19. Impagliazzo, R., Paturi, R., Zane, F.: Which problems have strongly exponential complexity? Journal of Computer and System Sciences 63(4), 512–530 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  20. Iswadi, H., Baskoro, E.T., Simanjuntak, R.: On the metric dimension of corona product of graphs. Far East Journal of Mathematical Sciences 52(2), 155–170 (2011)

    MATH  MathSciNet  Google Scholar 

  21. Jannesari, M., Omoomi, B.: The metric dimension of the lexicographic product of graphs. Discrete Mathematics 312(22), 3349–3356 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  22. Johnson, M.: Structure-activity maps for visualizing the graph variables arising in drug design. Journal of Biopharmaceutical Statistics 3(2), 203–236 (1993), pMID: 8220404

    Google Scholar 

  23. Johnson, M.A.: Browsable structure-activity datasets. In: Carbó-Dorca, R., Mezey, P. (eds.) Advances in Molecular Similarity, pp. 153–170. JAI Press Inc., Stamford (1998)

    Google Scholar 

  24. Karpovsky, M.G., Chakrabarty, K., Levitin, L.B.: On a new class of codes for identifying vertices in graphs. IEEE Transactions on Information Theory 44(2), 599–611 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  25. Khuller, S., Raghavachari, B., Rosenfeld, A.: Landmarks in graphs. Discrete Applied Mathematics 70, 217–229 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  26. Laifenfeld, M., Trachtenberg, A.: Identifying codes and covering problems. IEEE Transactions on Information Theory 54(9), 3929–3950 (2008)

    Article  MathSciNet  Google Scholar 

  27. Lichtenstein, D.: Planar formulae and their uses. SIAM Journal on Computing 11, 329–343 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  28. Lokshtanov, D., Marx, D., Saurabh, S.: Lower bounds based on the Exponential Time Hypothesis. EATCS Bulletin 105, 41–72 (2011)

    MATH  MathSciNet  Google Scholar 

  29. Okamoto, F., Phinezy, B., Zhang, P.: The local metric dimension of a graph. Mathematica Bohemica 135(3), 239–255 (2010)

    MATH  MathSciNet  Google Scholar 

  30. Rodríguez-Velázquez, J.A., Fernau, H.: On the (adjacency) metric dimension of corona and strong product graphs and their local variants: combinatorial and computational results. Tech. Rep. arXiv:1309.2275 [math.CO], ArXiv.org, Cornell University (2013)

    Google Scholar 

  31. Rodríguez-Velázquez, J.A., Barragán-Ramírez, G.A., Gómez, C.G.: On the local metric dimension of corona product graph (2013) (submitted)

    Google Scholar 

  32. Saputro, S., Simanjuntak, R., Uttunggadewa, S., Assiyatun, H., Baskoro, E., Salman, A., Bača, M.: The metric dimension of the lexicographic product of graphs. Discrete Mathematics 313(9), 1045–1051 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  33. Slater, P.J.: Leaves of trees. Congressus Numerantium 14, 549–559 (1975)

    MathSciNet  Google Scholar 

  34. Suomela, J.: Approximability of identifying codes and locating-dominating codes. Information Processing Letters 103(1), 28–33 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  35. Yero, I.G., Kuziak, D., Rodríquez-Velázquez, J.A.: On the metric dimension of corona product graphs. Computers & Mathematics with Applications 61(9), 2793–2798 (2011)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. FB 4-Abteilung Informatikwissenschaften, Universität Trier, 54286, Trier, Germany

    Henning Fernau

  2. Universitat Rovira i Virgili, Av. Països Catalans 26, 43007, Tarragona, Spain

    Juan Alberto Rodríguez-Velázquez

Authors
  1. Henning Fernau
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Juan Alberto Rodríguez-Velázquez
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Steklov Institute of Mathematics at St. Petersburg, Russian Academy of Sciences, 27 Fontanka,, 191023, St. Petersburg, Russia

    Edward A. Hirsch

  2. Higher School of Economics, National Research University, 109028, Moscow, Russia

    Sergei O. Kuznetsov

  3. CNRS and University Paris Diderot, Paris, France

    Jean-Éric Pin

  4. Moscow State University, 119992, Moscow, Russia

    Nikolay K. Vereshchagin

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this paper

Cite this paper

Fernau, H., Rodríguez-Velázquez, J.A. (2014). Notions of Metric Dimension of Corona Products: Combinatorial and Computational Results. In: Hirsch, E.A., Kuznetsov, S.O., Pin, JÉ., Vereshchagin, N.K. (eds) Computer Science - Theory and Applications. CSR 2014. Lecture Notes in Computer Science, vol 8476. Springer, Cham. https://doi.org/10.1007/978-3-319-06686-8_12

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-06686-8_12

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-06685-1

  • Online ISBN: 978-3-319-06686-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation