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The Local Metric Dimension of Strong Product Graphs

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Abstract

A vertex \(v\in V(G)\) is said to distinguish two vertices \(x,y\in V(G)\) of a nontrivial connected graph G if the distance from v to x is different from the distance from v to y. A set \(S\subset V(G)\) is a local metric generator for G if every two adjacent vertices of G are distinguished by some vertex of S. A local metric generator with the minimum cardinality is called a local metric basis for G and its cardinality, the local metric dimension of G. It is known that the problem of computing the local metric dimension of a graph is NP-Complete. In this paper we study the problem of finding exact values or bounds for the local metric dimension of strong product of graphs.

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References

  1. Bailey, R.F., Meagher, K.: On the metric dimension of grassmann graphs. Discrete Math. Theor. Comput. Sci. 13(4), 97–104 (2011). http://www.dmtcs.org/dmtcs-ojs/index.php/dmtcs/article/view/2049

  2. Barragán-Ramírez, G., Gómez, C.G., Rodríguez-Velázquez, J.A.: Closed formulae for the local metric dimension of corona product graphs. Electron. Notes Discrete Math. 46, 27–34 (2014). pii: S1571065314000067

  3. Blumenthal, L.M.: Theory and Applications of Distance Geometry, 2nd edn. Chelsea, New York (1970)

    MATH  Google Scholar 

  4. Brigham, R.C., Chartrand, G., Dutton, R.D., Zhang, P.: Resolving domination in graphs. Math. Bohem. 128(1), 25–36 (2003). http://mb.math.cas.cz/mb128-1/3.html

  5. Cáceres, J., Hernando, C., Mora, M., Pelayo, I.M., Puertas, M.L., Seara, C., Wood, D.R.: On the metric dimension of cartesian product of graphs. SIAM J. Discrete Math. 21(2), 423–441 (2007). doi:10.1137/050641867

    Article  MathSciNet  MATH  Google Scholar 

  6. Chartrand, G., Eroh, L., Johnson, M.A., Oellermann, O.R.: Resolvability in graphs and the metric dimension of a graph. Discrete Appl. Math. 105(1–3), 99–113 (2000). doi:10.1016/S0166-218X(00)00198-0

    Article  MathSciNet  MATH  Google Scholar 

  7. Chartrand, G., Saenpholphat, V., Zhang, P.: The independent resolving number of a graph. Math. Bohem. 128(4), 379–393 (2003). http://mb.math.cas.cz/mb128-4/4.html

  8. Estrada-Moreno, A., Rodríguez-Velázquez, J.A., Yero, I.G.: The \(k\)-metric dimension of a graph. Appl. Math. Inf. Sci. 9(6), 2829–2840 (2015). arXiv:1312.6840

  9. Feng, M., Wang, K.: On the metric dimension of bilinear forms graphs. Discrete Math. 312(6), 1266–1268 (2012). pii: S0012365X11005279

  10. Fernau, H., Rodríguez-Velázquez, J.A.: On the (adjacency) metric dimension of corona and strong product graphs and their local variants: combinatorial and computational results. arXiv:1309.2275 [math.CO], http://arxiv-web3.library.cornell.edu/abs/1309.2275

  11. Fernau, H., Rodríguez-Velázquez, J.A.: Notions of metric dimension of corona products: combinatorial and computational results. In: Computer Science—Theory and Applications, Lecture Notes in Computer Science, vol. 8476, pp. 153–166. Springer, Cham (2014)

  12. Guo, J., Wang, K., Li, F.: Metric dimension of some distance-regular graphs. J. Combin. Optim. 26, 190–197 (2013). doi:10.1007/s10878-012-9459-x

    Article  MathSciNet  MATH  Google Scholar 

  13. Hammack, R., Imrich, W., Klavžar, S.: Handbook of product graphs, Discrete Mathematics and its Applications, 2nd edn. CRC Press (2011). http://www.crcpress.com/product/isbn/9781439813041

  14. Harary, F., Melter, R.A.: On the metric dimension of a graph. Ars Combin. 2, 191–195 (1976). http://www.ams.org/mathscinet-getitem?mr=0457289

  15. Haynes, T.W., Henning, M.A., Howard, J.: Locating and total dominating sets in trees. Discrete Appl. Math. 154(8), 1293–1300 (2006). http://www.sciencedirect.com/science/article/pii/S0166218X06000035

  16. Johnson, M.: Structure-activity maps for visualizing the graph variables arising in drug design. J. Biopharm. Stat. 3(2), 203–236 (1993). http://www.tandfonline.com/doi/abs/10.1080/10543409308835060

  17. Johnson, M.: Browsable structure-activity datasets. In: Carbó-Dorca, R., Mezey, P. (eds.) Advances in Molecular Similarity, chap. 8, pp. 153–170. JAI Press Inc, Stamford (1998). http://books.google.es/books?id=1vvMsHXd2AsC

  18. Khuller, S., Raghavachari, B., Rosenfeld, A.: Landmarks in graphs. Discrete Appl. Math. 70(3), 217–229 (1996). http://www.sciencedirect.com/science/article/pii/0166218X95001062

  19. Melter, R.A., Tomescu, I.: Metric bases in digital geometry. Comput. Vis. Graphics Image Process. 25(1), 113–121 (1984). pii: 0734189X84900513

  20. Okamoto, F., Phinezy, B., Zhang, P.: The local metric dimension of a graph. Math. Bohem. 135(3), 239–255 (2010). http://dml.cz/dmlcz/140702

  21. Ramírez-Cruz, Y., Oellermann, O.R., Rodríguez-Velázquez, J.A.: Simultaneous resolvability in graph families. Electron. Notes Discrete Math. 46(0), 241–248 (2014). http://www.sciencedirect.com/science/article/pii/S157106531400033X

  22. Rodríguez-Velázquez, J.A., Barragán-Ramírez, G.A., García Gómez, C.: On the local metric dimension of corona product graphs. Bull. Malays. Math. Sci. Soc. (2015, to appear). http://arxiv-web3.library.cornell.edu/abs/1308.6689

  23. Rodríguez-Velázquez, J.A., García Gómez, C., Barragán-Ramírez, G.A.: Computing the local metric dimension of a graph from the local metric dimension of primary subgraphs. Int. J. Comput. Math. 92(4), 686–693 (2015). http://arxiv.org/abs/1402.0177

  24. Rodríguez-Velázquez, J.A., Kuziak, D., Yero, I.G., Sigarreta, J.M.: The metric dimension of strong product graphs. Carpathian J. Math. 31(2), 261–268 (2015)

    MathSciNet  MATH  Google Scholar 

  25. Saenpholphat, V., Zhang, P.: Conditional resolvability in graphs: a survey. Int. J. Math. Math. Sci. 2004(38), 1997–2017 (2004). http://www.hindawi.com/journals/ijmms/2004/247096/abs/

  26. Sebö, A., Tannier, E.: On metric generators of graphs. Math. Oper. Res. 29(2), 383–393 (2004). doi:10.1287/moor.1030.0070

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  28. Yero, I.G., Kuziak, D., Rodríquez-Velázquez, J.A.: On the metric dimension of corona product graphs. Comput. Math. Appl. 61(9), 2793–2798 (2011). pii: S0898122111002094

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Authors and Affiliations

  1. Departament d’Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, Av. Països Catalans 26, 43007, Tarragona, Spain

    Gabriel A. Barragán-Ramírez & Juan A. Rodríguez-Velázquez

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  1. Gabriel A. Barragán-Ramírez
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  2. Juan A. Rodríguez-Velázquez
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Correspondence to Juan A. Rodríguez-Velázquez.

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Barragán-Ramírez, G.A., Rodríguez-Velázquez, J.A. The Local Metric Dimension of Strong Product Graphs. Graphs and Combinatorics 32, 1263–1278 (2016). https://doi.org/10.1007/s00373-015-1653-z

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