Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

1 Issue per year


IMPACT FACTOR 2016 (Open Mathematics): 0.682
IMPACT FACTOR 2016 (Central European Journal of Mathematics): 0.489

CiteScore 2016: 0.62

SCImago Journal Rank (SJR) 2016: 0.454
Source Normalized Impact per Paper (SNIP) 2016: 0.850

Mathematical Citation Quotient (MCQ) 2016: 0.23

Open Access
Online
ISSN
2391-5455
See all formats and pricing
More options …
Volume 13, Issue 1 (Jan 2015)

Issues

On the strong metric dimension of the strong products of graphs

Dorota Kuziak
  • Departament d’Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, Av. Països Catalans 26, 43007 Tarragona, Spain
  • Email:
/ Ismael G. Yero
  • Corresponding author
  • Departamento de Matemáticas, Escuela Politécnica Superior de Algeciras, Universidad de Cádiz, Av. Ramón Puyol s/n, 11202 Algeciras, Spain
  • Email:
/ Juan A. Rodríguez-Velázquez
  • Corresponding author
  • Departament d’Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, Av. Països Catalans 26, 43007 Tarragona, Spain
  • Email:
Published Online: 2014-10-09 | DOI: https://doi.org/10.1515/math-2015-0007

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.

Keywords : Strong metric dimension; Strong metric basis; Strong resolving set; Strong product 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 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 Web of ScienceGoogle 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 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 CrossrefWeb of ScienceGoogle Scholar

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

  • [7] Hales R. S., Numerical invariants and the strong product of graphs, J. Combin. Theory Ser. B, 1973, 15, 146–155 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 Google Scholar

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

  • [10] Jannesari M., Omoomi B., The metric dimension of the lexicographic product of graphs, Discrete Math. 2012, 312(22), 3349–3356 Google Scholar

  • [11] Jha P. K., Slutzki G., Independence numbers of product graphs, Appl. Math. Lett., 1994, 7(4), 91–94 CrossrefGoogle Scholar

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

  • [14] Khuller S., Raghavachari B., Rosenfeld A., Landmarks in graphs, Discrete Appl. Math., 1996, 70, 217–229 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 Web of ScienceCrossrefGoogle 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 Web of ScienceGoogle Scholar

  • [17] Melter R. A., Tomescu I., Metric bases in digital geometry, Computer Vision, Graphics, and Image Processing, 1984, 25, 113–121 Google Scholar

  • [18] Oellermann O. R., Peters-Fransen J., The strong metric dimension of graphs and digraphs, Discrete Appl. Math., 2007, 155, 356–364 Web of ScienceGoogle Scholar

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

  • [23] Seb˝o A., Tannier E., On metric generators of graphs, Math. Oper. Res., 2004, 29(2), 383–393 CrossrefGoogle 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 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), 2793–2798Web of ScienceGoogle Scholar

About the article

Received: 2013-08-28

Accepted: 2014-07-11

Published Online: 2014-10-09

Published in Print: 2015-01-01


Citation Information: Open Mathematics, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2015-0007.

Export Citation

© 2015 Dorota Kuziak et al.. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

Comments (0)

Please log in or register to comment.
Log in