Abstract
Let \(G=(V, E)\) be a connected graph. Given a vertex \(v\in V\) and an edge \(e=uw\in E\), the distance between v and e is defined as \(d_G(e,v)=\min \{d_G(u,v),d_G(w,v)\}\). A nonempty set \(S\subset V\) is an edge metric generator for G if for any two edges \(e_1,e_2\in E\) there is a vertex \(w\in S\) such that \(d_G(w,e_1)\ne d_G(w,e_2)\). The minimum cardinality of any edge metric generator for a graph G is the edge metric dimension of G. The edge metric dimension of the join, lexicographic, and corona product of graphs is studied in this article.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
The graph \(G\vee K_1\) is obtained from a graph G and a vertex v, by joining with and edge every vertex of G with the vertex v.
References
Barragán-Ramírez, G.A., Estrada-Moreno, A., Ramírez-Cruz, Y., Rodríguez-Velázquez, J.A.: The local metric dimension of the lexicographic product of graphs. Bull. Malays. Math. Sci. Soc. 42(5), 2481–2496 (2019)
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)
Estrada Moreno, A.: On the \((k,t)\)-Metric Dimension of a Graph, Ph. D. dissertation. Universitat Rovira i Virgili (2016)
Estrada-Moreno, A., Yero, I.G., Rodríguez-Velázquez, J.A.: The \(k\)-metric dimension of the lexicographic product of graphs. Discrete Math. 339(7), 1924–1934 (2016)
Geneson, J.: Metric dimension and pattern avoidance in graphs, manuscript. arXiv:1807.08334 [math.CO] (2018)
Hammack, R., Imrich, W., Klavžar, S.: Handbook of Product Graphs, 2nd edn. CRC Press, Boca Raton (2011)
Harary, F., Melter, R.A.: On the metric dimension of a graph. Ars Combin. 2, 191–195 (1976)
Hernando, C., Mora, M., Pelayo, I.M., Seara, C., Wood, D.R.: Extremal graph theory for metric dimension and diameter. Electron. J. Combin. 17(1), 30 (2010)
Jannesari, M., Omoomi, B.: The metric dimension of the lexicographic product of graphs. Discrete Math. 312(22), 3349–3356 (2012)
Kelenc, A., Kuziak, D., Taranenko, A., Yero, I.G.: On the mixed metric dimension of graphs. Appl. Math. Comput. 314, 429–438 (2017)
Kelenc, A., Tratnik, N., Yero, I.G.: Uniquely identifying the edges of a graph: the edge metric dimension. Discrete Appl. Math. 251, 204–220 (2018)
Kuziak, D.: Strong resolvability in product graphs, Ph. D. dissertation, Universitat Rovira i Virgili (2014)
Ramírez Cruz, Y.: The simultaneous (strong) metric dimension of graph families, Ph. D. dissertation, Universitat Rovira i Virgili (2016)
Slater, P.J.: Leaves of trees. Congr. Numer. 14, 549–559 (1975)
West, D.B.: Introduction to Graph Theory, 2nd edn. Prentice Hall, Upper Saddle River (2001)
Yero, I.G., Kuziak, D., Rodríguez-Velázquez, J.A.: On the metric dimension of corona product graphs. Comput. Math. Appl. 61(9), 2793–2798 (2011)
Zubrilina, N.: On the edge dimension of a graph. Discrete Math. 341, 2083–2088 (2018)
Author information
Authors and Affiliations
Corresponding author
Additional information
Sandi Klavžar.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Peterin, I., Yero, I.G. Edge Metric Dimension of Some Graph Operations. Bull. Malays. Math. Sci. Soc. 43, 2465–2477 (2020). https://doi.org/10.1007/s40840-019-00816-7
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40840-019-00816-7