Metric Dimension of Shackle Operation C_3 Cycle Graph

St. Munieroh   Fachrunnisa; Hasmawati Hasmawati; Amir Kamal  Amir

doi:10.20956/j.v19i2.22957

Authors

St. Munieroh Fachrunnisa Universitas Hasanuddin
Hasmawati Hasmawati
Amir Kamal Amir

DOI:

https://doi.org/10.20956/j.v19i2.22957

Keywords:

Metric Dimension, Shackle Operation, Cycle Graph

Abstract

Let G be a connected graph and W be a ordered vertices subset on a connected graph . The set W is called resolving set for G if every vertex on graph G has distinct representation of W. A resolving set containing a minimum number of vertices is called resolving set minimum or basis for G and the cardinality of resolving set is the metric dimension on graph G, denoted by dim(G). In the thesis discusses about metric dimensions of shackle operation C₃ cycle graph, dim(Shack(C₃¹,C₃²,…,C₃^k:v₃¹=v₁²,v₃²=v₁³,…,v₃^k-1=v₁^k ))=2 for k>=2 . To proof this results, we was used mathematical induction method.

References

Chartrand, G., Eroh, L., Johnson, M. A., & Oellermann, O. R., 2000. Resolvability in graphs and the metric dimension of a graph. Discrete Applied Mathematics, Vol. 105, 99–113.

Eka R, S. & Rahadjeng, B., 2014. Dimensi Metrik pada Graf Lintasan, Graf Komplit, Graf Sikel, Graf Bintang, dan Graf Bipartit Komplit. Jurnal : Universitas Negeri Surabaya, Vol. 1, No. 1, 1-6.

Harary, F. & Melter, R., 1976. On the Metric Dimension of a Graph. Ars Combin. Vol. 2 : 191-195.

Hasmawati, 2020. Pengantar dan Jenis-Jenis Graf. UPT Unhas Press, Makassar.

Hasmawati, Hiding, N., Nurwahyu, B., Syukur Daming, A., & Kamal Amir, A., 2022. The partition dimension of the vertex amalgamation of some cycles. Heliyon, Vol. 8, 1-7.

Liu, J., Faisal Nadeem, M., Muhammad Afzal Siddiqui, H., & Nazir, W., 2019. Computing Metric Dimension of Certain Families of Toeplitz Graphs. IEEE Access, Vol. 7, 126734-126741.

Marsidi., Hesti Agustin, I., Dafik., Alfarisi, R., & Siswono, H., 2018. On the Metric Dimension of Some Operation Graphs. Cauchy-Jurnal Matematika Murni dan Aplikasi, Vol. 5, No. 3, 88-94.

Rezaei, A., Khashyarmanesha, K., & Afkhamib, M., 2022. On the metric dimension of Cayley graphs. AKCE International Journal of Graphs and Combinatorics, Vol. 19, No. 2, 118-124.

Saifuddin, I., 2015. Dimensi Partisi dari Graf Khusus dan Operasinya. Skripsi. FMIPA Universitas Jember, Jember.

Saifuddin, I., Umilasari, R., & Jalil, A., 2021. Metric dimension of vertex shackle operation result graph on wheel and star graphs. Journal of Physics: Conference Series, Vol. 2279, No. 1, 1-5.

Wahyudi, S., Sumarno, & Suharmadi, 2011. Dimensi Metrik Pengembangan Graf Kincir Pola ?1+??3. Journal mathematics and its applied, Vol. 8, No. 2, 17-22.