Advertisement

On the fault-tolerant metric dimension of certain interconnection networks

  • Hassan Raza
  • Sakander Hayat
  • Xiang-Feng PanEmail author
Original Research

Abstract

Metric dimension and fault-tolerant metric dimension have potential applications in telecommunication, robot navigation and geographical routing protocols, among others. The computational complexity of these problems is known to be NP-complete. In this paper, we study the fault-tolerant metric dimension of various interconnection networks. By using the resolving sets in these networks, we locate fault-tolerant resolving sets in them. As a result, certain lower and upper bounds on the fault-tolerant metric dimension of those networks are obtained. We conclude the paper with some open problems.

Keywords

Metric dimension Fault-tolerant metric dimension Grid networks Hexagonal networks Honeycomb network Hex-derived networks 

Mathematics Subject Classification

05C12 05C76 05C90 

Notes

Acknowledgements

The authors are grateful to the anonymous reviewers for a careful reading of this paper and for all their comments, which lead to a number of improvements of the paper.

References

  1. 1.
    Ahmad, A., Imran, M., Al-Mushayt, O., Bokhary, S.A.U.H.: On the metric dimension of barycentric subdividion of Cayley graph \(Cay(\mathbb{Z}_n\oplus \mathbb{Z}_m)\). Miskolc. Math. Notes 16(2), 637–646 (2015)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bailey, R.F., Cameron, P.J.: Basie size, metric dimension and other invariants of groups and graphs. Bull. London Math. Soc. 43, 209–242 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bailey, R.F., Meagher, K.: On the metric dimension of Grassmann graphs. Discrete Math. Theor. Comput. Sci. 13(4), 97–104 (2011)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Beerloiva, Z., Eberhard, F., Erlebach, T., Hall, A., Hoffmann, M., Mihalák, M., Ram, L.: Network discovery and verification. IEEE J. Sel. Area Commun. 24, 2168–2181 (2006)CrossRefGoogle Scholar
  5. 5.
    Bondy, J.A., Murty, U.S.R.: Graph Theory. Springer, New York (2008)CrossRefzbMATHGoogle Scholar
  6. 6.
    Cáceres, J., Hernando, C., Mora, M., Pelayoe, I.M., Puertas, M.L.: On the metric dimension of infinite graphs. Electron. Notes Discrete Math. 35, 15–20 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cáceres, J., Hernando, C., Mora, M., Pelayoe, 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)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chartrand, G., Zhang, P.: The theory and applications of resolvability in graphs: A survey. In: Proceedings of the 34th Southeastern International Conference on Combinatorics, Graph Theory and Computing. Congr. Numer. 160, 47–68 (2003)Google Scholar
  9. 9.
    Chartrand, G., Eroh, L., Johnson, M.A., Oellermann, O.R.: Resolvability in graphs and the metric dimension of a graph. Discrete Appl. Math. 150, 99–113 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Chen, M.S., Shin, K.G., Kandlur, D.D.: Addressing, routing and broadcasting in hexagonal mesh multiprocessors. IEEE Trans. Comput. 39, 10–18 (1990)CrossRefGoogle Scholar
  11. 11.
    Fehr, M., Gosselin, S., Oellermann, O.: The metric dimension of Cayley digraphs. Discrete Math. 306, 31–41 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman and Company, New York (1979)zbMATHGoogle Scholar
  13. 13.
    Harary, F., Melter, R.A.: On the metric dimension of a graph. Ars Combin. 2, 191–195 (1976)zbMATHGoogle Scholar
  14. 14.
    Hayat, S.: Computing distance-based topological descriptors of complex chemical networks: new theoretical techniques. Chem. Phys. Lett. 688, 51–58 (2017)CrossRefGoogle Scholar
  15. 15.
    Hayat, S., Malik, M.A., Imran, M.: Computing topological indices of honeycomb derived networks. Rom. J. Inf. Sci. Technol. 18, 144–165 (2015)Google Scholar
  16. 16.
    Hernando, C., Mora, M., Slater, P.J., Wood, D.R.: Fault-tolerant metric dimension of graphs. In: Proceedings of International Conference on Convexity in Discrete Structures, Ramanujan Mathematical Society Lecture Notes, pp. 81–85. May (2008)Google Scholar
  17. 17.
    Imran, M., Siddiqui, H.M.A.: Computing the metric dimension of convex polytopes generated by the wheel related graphs. Acta Math. Hung. 149, 10–30 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Javaid, I., Salman, M., Chaudhry, M.A., Shokat, S.: Fault-tolerance in resolvibility. Utilitas Math. 80, 263–275 (2009)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Khuller, S., Raghavachari, B., Rosenfeld, A.: Landmarks in graphs. Discrete Appl. Math. 70, 217–229 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Kratica, J., Kovačević-Vujčić, V., Čangalović, M., Stojanović, M.: Minimal doubly resolving sets and the strong metric dimension of some convex polytopes. Appl. Math. Comput. 218, 9790–9801 (2012)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Krishnan, S., Rajan, B.: Fault-tolerant resolvability of certain crystal structures. Appl. Math. 7, 599–604 (2016)CrossRefGoogle Scholar
  22. 22.
    Lester, L.N., Sandor, J.: Computer graphics on hexagonal grid. Comput. Graph. 8, 401–409 (1984)CrossRefGoogle Scholar
  23. 23.
    Liu, K., Abu-Ghazaleh, N.: Virtual coordinate back tracking for void travarsal in geographic routing. In: Kunz, T., Ravi, S.S. (eds.) Ad-Hoc, Mobile, and Wireless Networks. ADHOC-NOW 2006. Lecture Notes in Computer Science, vol. 4104. Springer, Berlin (2006)Google Scholar
  24. 24.
    Manuel, P., Rajan, B., Rajasingh, I., Monica, C.: On minimum metric dimension of honeycomb networks. J. Discrete Algorithms 6, 20–27 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Nocetti, F.G., Stojmenovic, I., Zhang, J.: Addressing and routing in hexagonal networks with applications for tracking mobile users and connection rerouting in cellular networks. IEEE Trans. Parallel Distrib. Syst. 13, 963–971 (2002)CrossRefGoogle Scholar
  26. 26.
    Parhami, B., Kwai, D.-M.: A unified formulation of honeycomb and diamond networks. IEEE Trans. Parallel Distrib. Syst. 12, 74–79 (2001)CrossRefGoogle Scholar
  27. 27.
    Raza, H., Hayat, S., Pan, X.-F.: On the fault-tolerant metric dimension of convex polytopes. Appl. Math. Comput. 339, 172–185 (2018)MathSciNetGoogle Scholar
  28. 28.
    Salman, M., Javaid, I., Chaudhry, M.A.: Minimum fault-tolerant, local and strong metric dimension of graphs, arXiv preprint arXiv:1409.2695 (2014), http://arxiv.org/pdf/1409.2695
  29. 29.
    Siddiqui, H.M.A., Imran, M.: Computing the metric dimension of wheel related graphs. Appl. Math. Comput. 242, 624–632 (2014)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Slater, P.J.: Leaves of trees. In: Proceedings of 6th Southeastern Conference on Combinatorics, Graph Theory, and Computing. Congr. Numer. 549–559 (1975)Google Scholar
  31. 31.
    Stojmenovic I.: Direct interconnection networks. In: Zomaya, A.Y. (ed.) Parallel and Distributed Computing Handbook, McGraw-Hill Professional, pp. 537–567 (1996)Google Scholar
  32. 32.
    Stojmenovic, I.: Honeycomb networks: topological properties and communication algorithms. IEEE Trans. Parallel Distrib. Syst. 8, 1036–1042 (1997)CrossRefGoogle Scholar
  33. 33.
    Vetrík, T., Ahmad, A.: Computing the metric dimension of the categorial product of some graphs. Int. J. Comput. Math. 94(2), 363–371 (2015)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Korean Society for Computational and Applied Mathematics 2018

Authors and Affiliations

  1. 1.School of Mathematical SciencesAnhui UniversityHefeiPeople’s Republic of China
  2. 2.Faculty of Engineering SciencesGIK Institute of Engineering Sciences and TechnologyTopi, SwabiPakistan

Personalised recommendations