Skip to content
BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access October 9, 2014

On the strong metric dimension of the strong products of graphs

  • Dorota Kuziak , Ismael G. Yero EMAIL logo and Juan A. Rodríguez-Velázquez EMAIL logo
From the journal Open Mathematics

Abstract

Let G be a connected graph. A vertex w ∈ V.G/ strongly resolves two vertices u,v ∈ V.G/ if there exists some shortest u-w path containing v or some shortest v-w path containing u. A set S of vertices is a strong resolving set for G if every pair of vertices of G is strongly resolved by some vertex of S. The smallest cardinality of a strong resolving set for G is called the strong metric dimension of G. It is well known that the problem of computing this invariant is NP-hard. In this paper we study the problem of finding exact values or sharp bounds for the strong metric dimension of strong product graphs and express these in terms of invariants of the factor graphs.

References

[1] Cáceres J. , Hernando C., Mora M., Pelayo I. M., Puertas M. L., Boundary-type sets and product operators in graphs, In: VII Jornadas de Matemática Discreta y Algorítmica, Castro Urdiales, Cantabria, Spain, July 2010 Search in Google Scholar

[2] 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., 2007, 21(2), 273–302 10.1137/050641867Search in Google Scholar

[3] Cˇ ižek N., Klavžar S., On the chromatic number of the lexicographic product and the Cartesian sum of graphs, Discrete Math., 1994, 134(1-3), 17–24 10.1016/0012-365X(93)E0056-ASearch in Google Scholar

[4] Feng M., Wang K., On the metric dimension and fractional metric dimension of the hierarchical product of graphs, Appl. Anal. Discrete Math., 2013, 7, 302–313 10.2298/AADM130521009FSearch in Google Scholar

[5] Gallai T., Uber Extreme Punkt-und Kantenmengen, Ann. Univ. Sci. Budapest Eötvös Sect. Math., 1959, 2, 133–138 (in German) Search in Google Scholar

[6] Geller D., Stahl S., The chromatic number and other functions of the lexicographic product, J. Combin. Theory Ser. B, 1975, 19, 87–95 10.1016/0095-8956(75)90076-3Search in Google Scholar

[7] Hales R. S., Numerical invariants and the strong product of graphs, J. Combin. Theory Ser. B, 1973, 15, 146–155 10.1016/0095-8956(73)90014-2Search in Google Scholar

[8] Hammack R., Imrich W., Klavžar S., Handbook of Product Graphs, Second edition. With a foreword by Peter Winkler. Discrete Math. Appl. (Boca Raton), CRC Press, Boca Raton, FL, 2011 10.1201/b10959Search in Google Scholar

[9] Harary F., Melter R. A., On the metric dimension of a graph, Ars Combin., 1976, 2, 191–195 Search in Google Scholar

[10] Jannesari M., Omoomi B., The metric dimension of the lexicographic product of graphs, Discrete Math. 2012, 312(22), 3349–3356 10.1016/j.disc.2012.07.025Search in Google Scholar

[11] Jha P. K., Slutzki G., Independence numbers of product graphs, Appl. Math. Lett., 1994, 7(4), 91–94 10.1016/0893-9659(94)90018-3Search in Google Scholar

[12] Johnson M. A., Structure-activity maps for visualizing the graph variables arising in drug design, J. Biopharm. Statist., 1993, 3, 203–236 10.1080/10543409308835060Search in Google Scholar

[13] Johnson M. A., Browsable structure-activity datasets, In: Advances in Molecular Similarity, R. Carbó–Dorca and P. Mezey, eds., JAI Press Connecticut, 1998, 153–170 10.1016/S1873-9776(98)80014-XSearch in Google Scholar

[14] Khuller S., Raghavachari B., Rosenfeld A., Landmarks in graphs, Discrete Appl. Math., 1996, 70, 217–229 10.1016/0166-218X(95)00106-2Search in Google Scholar

[15] Kratica J., Kovacˇevic´-Vujcˇic´ V., Cˇ angalovic´ M., Stojanovic´ M., Minimal doubly resolving sets and the strong metric dimension of Hamming graphs, Appl. Anal. Discrete Math., 2012, 6, 63–71 10.2298/AADM111116023KSearch in Google Scholar

[16] Kuziak D., Yero I. G., Rodríguez-Velázquez J. A., On the strong metric dimension of corona product graphs and join graphs, Discrete Appl. Math., 2013, 161(7-8), 1022–1027 10.1016/j.dam.2012.10.009Search in Google Scholar

[17] Melter R. A., Tomescu I., Metric bases in digital geometry, Computer Vision, Graphics, and Image Processing, 1984, 25, 113–121 10.1016/0734-189X(84)90051-3Search in Google Scholar

[18] Oellermann O. R., Peters-Fransen J., The strong metric dimension of graphs and digraphs, Discrete Appl. Math., 2007, 155, 356–364 10.1016/j.dam.2006.06.009Search in Google Scholar

[19] Ore, O., Theory of graphs, Amer. Math. Soc. Colloq. Publ., 38, Amer. Math. Soc., Providence, R.I., 1962 Search in Google Scholar

[20] Rodríguez-Velázquez J. A., Kuziak D., Yero I. G., Sigarreta J. M., The metric dimension of strong product graphs, Carpathian J. Math. (in press), preprint available at http://arxiv.org/abs/1305.0363 Search in Google Scholar

[21] Saputro S., Simanjuntak R., Uttunggadewa S., Assiyatun H., Baskoro E., Salman A., Baˇca M., The metric dimension of the lexicographic product of graphs, Discrete Math., 2013, 313(9), 1045–1051 10.1016/j.disc.2013.01.021Search in Google Scholar

[22] Scheinerman E., Ullman D., Fractional Graph Theory. A rational approach to the theory of graphs. With a foreword by Claude Berge, Wiley-Intersci. Ser. Discrete Math. Optim., A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1997 Search in Google Scholar

[23] Seb˝o A., Tannier E., On metric generators of graphs, Math. Oper. Res., 2004, 29(2), 383–393 10.1287/moor.1030.0070Search in Google Scholar

[24] Slater P. J., Leaves of trees, In: Proceeding of the 6th Southeastern Conference on Combinatorics, Graph Theory, and Computing. Congr. Numer., 1975, 14, 549–559 Search in Google Scholar

[25] Yero I. G., Kuziak D., Rodríguez-Velázquez J. A., On the metric dimension of corona product graphs, Comput. Math. Appl., 2011, 61(9), 2793–279810.1016/j.camwa.2011.03.046Search in Google Scholar

Received: 2013-8-28
Accepted: 2014-7-11
Published Online: 2014-10-9
Published in Print: 2015-1-1

© 2015 Dorota Kuziak et al.

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Downloaded on 6.12.2023 from https://www.degruyter.com/document/doi/10.1515/math-2015-0007/html
Scroll to top button