On The Local Metric Dimension of Line Graph of Special Graph

Marsidi Marsidi, Dafik Dafik, Ika Hesti Agustin, Ridho Alfarisi


Let G be a simple, nontrivial, and connected graph.  is a representation of an ordered set of k distinct vertices in a nontrivial connected graph G. The metric code of a vertex v, where , the ordered  of k-vector is representations of v with respect to W, where  is the distance between the vertices v and wi for 1≤ i ≤k.  Furthermore, the set W is called a local resolving set of G if  for every pair u,v of adjacent vertices of G. The local metric dimension ldim(G) is minimum cardinality of W. The local metric dimension exists for every nontrivial connected graph G. In this paper, we study the local metric dimension of line graph of special graphs , namely path, cycle, generalized star, and wheel. The line graph L(G) of a graph G has a vertex for each edge of G, and two vertices in L(G) are adjacent if and only if the corresponding edges in G have a vertex in common.


metric dimension, local metric dimension number, line graph, resolving set

Full Text:



Ulfianita, E., Estuningsih, N., Susilowati, L. Dimensi Metrik Lokal pada Graf Hasil Kali Comb dari Graf Siklus dan Graf Lintasan. Journal of Mathematics, Vol. 1 (2014) No. 3, 24 - 33.

Harary, F dan Melter, R. A. 1976. On The Metric Dimension of Graph. Ars Combin. 2(1):191-195.

Okamoto, F., Phinezy, B., Zhang, P. The Local Metric Dimension Of A Graph. Mathematica Bohemica, Vol. 135 (2010) No. 3, 239 - 255.

Gross, J.L., Yellen, J., Zhang P., Handbook of Graph Theory, Second Edition, CRC Press, Taylor and Francis Group, 2014.

Chartrand, G., Eroh, L., Johnson, M.A., & Oellermann, O.R., 2000, Resolvability in Graphs and the Metric Dimension of a Graph, Discrete Appl. Math., 105: 99-113.

Saputro, S.W., Suprijanto, D., Baskoro, E.T., & Salman, A. N. M., 2012, The Metric Dimension of a Graph Composition Products With Star, J. Indones. Math. Soc., Vol. 18 No. 2: 85-92.

Iswadi, H., Baskoro, E.T., & Simanjuntak, R., 2011, On the Metric Dimension of Corona Product Graphs, Far East Journal of Mathematical Sciences (FJMS),52 (2): 155 170.

Rodriguez-Velazquez, J.A. & Fernau, H. 2013, On the (Adjacency) Metric Dimension of Corona and Strong Product Graph and Their Local Variants: Combinatorial and Computational results, Combinatorial And Computational Results, Arxiv: 1309.2275.v1 (Math Co), 9 September.

Baca, M., Baskoro, E.T., Salman, A. N. M., Saputro, S. W., & Suprijanto, D., 2011. The Metric Dimension of Regular Bipartite Graphs, Bull. Math. Soc. Sci. Math. Roumanie, Tome 54(102) No. 1: 15-28.

Yero, I.G., Kuziak, D., & Rodriguez-Velazquez, J.A. , 2010, On the Metric Dimension of Corona Product Graphs, Combinatorial and Computational Results, arXiv:1009.2586v2[math CO] 7 Oct.

Permana, A. B., and Darmaji, Dimensi Metrik Graf Pohon Bentuk Tertentu, Jurnal Teknik Pomits Vol. 1, No. 1, (2012) 1-4 .

DOI: https://doi.org/10.18860/ca.v4i3.3694


  • There are currently no refbacks.

Copyright (c) 2016 Marsidi Marsidi, Dafik Dafik, Ika Hesti Agustin, Ridho Alfarisi

Creative Commons License
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

Editorial Office
Mathematics Department,
Universitas Islam Negeri Maulana Malik Ibrahim Malang
Gajayana Street 50 Malang, East Java, Indonesia 65144
Faximile (+62) 341 558933
e-mail: cauchy@uin-malang.ac.id

Creative Commons License
CAUCHY: Jurnal Matematika Murni dan Aplikasi is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.