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Conditional Resolvability of Honeycomb and Hexagonal Networks

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Abstract

Given a graph G = (V, E), a set \({W \subseteq V}\) is said to be a resolving set if for each pair of distinct vertices \({u, v \in V}\) there is a vertex x in W such that \({d(u, x) \neq d(v, x)}\) . The resolving number of G is the minimum cardinality of all resolving sets. In this paper, conditions are imposed on resolving sets and certain conditional resolving parameters are studied for honeycomb and hexagonal networks.

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References

  1. András S., Eric T.: On metric generators of graphs. Math. Oper. Res. 29(2), 383–393 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. Beerliova Z., Eberhard F., Erlebach T., Hall A., Hoffman M., Mihalák M.: Network discovery and verification. IEEE J. Sel. Areas Commun. 24(12), 2168–2181 (2006)

    Article  Google Scholar 

  3. Bharati R., Indra R., Cynthia J.A., Paul M.: On Minimum Metric Dimension. In: Proceedings of the Indonesia-Japan Conference on Combinatorial Geometry and Graph Theory, September 13–16, Bandung, Indonesia (2003)

  4. Bharati R., Indra R., Chris Monica M., Paul M.: Metric dimension of enhanced hypercube networks. J. Combin. Math. Combin. Comput. 67, 5–15 (2008)

    MathSciNet  MATH  Google Scholar 

  5. Bharati R., Indra R., Venugopal P., Chris Monica M.: Minimum metric dimension of illiac networks. Ars Combin. (Accepted)

  6. Chartrand G., Eroh L., Johnson M.A., Oellermann O.: Resolvability in graphs and the metric dimension of a graph. Discret. Appl. Math. 105, 99–113 (2000)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  8. Goddard W.: Statistic mastermind revisited. J. Combin. Math. Combin. Comput. 51, 215–220 (2004)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

  10. Johnson M.A.: Structure-activity maps for visualizing the graph variables arising in drug design. J. Biopharm. Stat. 3, 203–236 (1993)

    Article  MATH  Google Scholar 

  11. Khuller S., Ragavachari B., Rosenfield A.: Landmarks in Graphs. Discret. Appl. Math. 70(3), 217–229 (1996)

    Article  MATH  Google Scholar 

  12. Melter R.A., Tomcscu I.: Metric bases in digital geometry. Comput. Vis. Graph. Image Process. 25, 113–121 (1984)

    Article  Google Scholar 

  13. Paul M., Abd-El-Barr M.I., Indra R., Bharati R.: An Efficient Representation of Benes Networks and its Applications. J. Discret. Algorithms 6(1), 11–19 (2008)

    Article  MATH  Google Scholar 

  14. Paul M., Bharati R., Indra R., Chris Monica M.: On minimum metric dimension of honeycomb networks. J. Discret. Algorithms 6(1), 20–27 (2008)

    Article  MATH  Google Scholar 

  15. Paul M., Bharati R., Indra R., Chris Monica M.: Land marks in binary tree derived architectures. Ars Combin. 99, 473–486 (2011)

    MathSciNet  Google Scholar 

  16. Paul M., Bharati R., Indra R., Chris Monica M.: Landmarks in Torus Networks. J. Discret. Math. Sci. Cryptograph. 9(2), 263–271 (2006)

    MATH  Google Scholar 

  17. Saenpholphat V., Zhang P.: Conditional resolvability of graphs: a survey. IJMMS 38, 1997–2017 (2003)

    Google Scholar 

  18. Slater P.J.: Leaves of trees. Congr. Numer. 14, 549–559 (1975)

    MathSciNet  Google Scholar 

  19. Slater P.J.: Dominating and reference sets in a graph. J. Math. Phys. Sci. 22(4), 445–455 (1988)

    MathSciNet  MATH  Google Scholar 

  20. Sharieh A., Qatawneh M., Almobaideen W., Sleit A.: Hex-Cell: modeling, topological properties and routing algorithm. Eur. J. Sci. Res. 22(2), 457–468 (2008)

    Google Scholar 

  21. Söderberg S., Shapiro H. S.: A combinatory detection problem. Am. Math. Monthly 70, 1066 (1963)

    Article  MATH  Google Scholar 

  22. Stojmenovic I.: Networks: topological properties and communication algorithms. IEEE Trans. Parallel Distrib. Syst. 8, 1036–1042 (1997)

    Article  Google Scholar 

  23. Trobec R.: Two-dimensional regular d-meshes. Parallel Comput. 26, 1945–1953 (2000)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Loyola College, Chennai, 600 034, India

    B. Rajan, K. T. Sonia & M. Chris Monica

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  1. B. Rajan
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  2. K. T. Sonia
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  3. M. Chris Monica
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Correspondence to B. Rajan.

Additional information

This research is supported by The Major Research Project- No. F. 38-120/2009(SR) of the University Grants Commission, New Delhi, India.

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Rajan, B., Sonia, K.T. & Chris Monica, M. Conditional Resolvability of Honeycomb and Hexagonal Networks. Math.Comput.Sci. 5, 89–99 (2011). https://doi.org/10.1007/s11786-011-0076-3

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  • DOI: https://doi.org/10.1007/s11786-011-0076-3

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