Gold Open Access

Star metric dimension of complete, bipartite, complete bipartite and fan graphs

Reni Umilasari(1Mail), Ilham Saifudin(2), Isnawati Lujeng Lestari(3),
(1) Department of Informatics Engineering, Universitas Muhammadiyah Jember, Jawa Timur, 68121, Indonesia
(2) Department of Informatics Engineering, Universitas Muhammadiyah Jember, Jawa Timur, 68121, Indonesia
(3) Department of Mathematics Education, Institut Teknologi dan Sains Nahdlatul Ulama Pasuruan, Jawa Timur, 67171, Indonesia

Mail Corresponding Author

Abstract


One of the topics in graph theory that is interesting and developed continuously is metric dimension.  It has some new variation concepts, such as star metric dimension. The order set of Z={z1,z2,…,zn }⊆V(G) called star resolving set of connected graph  G if Z is Star Graph and for every vertex in G has different representation to the set Z. The representation is expressed as the distance d(u,z), it is the shortest path from vertex u to z for every u,z∈V(G). Star basis of a graph is the smallest cardinality of star resolving set. The number of vertex in star basis is called star metric dimension of G which denoted by Sdim(G). The purpose of this article is to determine the characteristic of star metric dimension and the value of star metric dimension of some classes of graphs. The method which is used in this study is library research. Some of the results of this research are complete graph has  Sdim(Kn )=n-1, for n≥3, bipartite graph K(2,n)  has  Sdim (K(2,n) )=n, for n≥3. Besides complete bipartite graph hasn’t star metric dimension or  for m,n≥3 or it can said that  Sdim (K(m,n) )=0. Another graph, that is Fan graph has Sdim (Fn )=2 for 2≤n≤5 and for n≥6 Sdim (Fn )=(2n+3)/5.

 


Keywords


metric dimension; star basis; star metric dimension; star resolving set; complete bipartite;

   

Article DOI



DOI: https://doi.org/10.33122/ijtmer.v5i2.137 		       

Article Metrics

 Abstract views : 822 
 PDF views : 289 


   

Article Pages


Pages: 199-205

   

Full Text:

PDF

References


Kang, C. X., Yero, I. G., & Yi, E. (2018). The fractional strong metric dimension in three graph products. Discrete Applied Mathematics, 251, 190–203. https://doi.org/10.1016/j.dam.2018.05.051

Kuziak, D., Puertas, M. L., Rodríguez-Velázquez, J. A., & Yero, I. G. (2018). Strong resolving graphs: The realization and the characterization problems. Discrete Applied Mathematics, 236, 270–287. https://doi.org/10.1016/j.dam.2017.11.013

Kuziak, D., Rodriguez-Velazquez, J. A., & Yero, I. G. (2017). Computing the metric dimension of a graph from primary subgraphs. Discussiones Mathematicae - Graph Theory, 37(1), 273–293. https://doi.org/10.7151/dmgt.1934

Kuziak, D., Yero, I. G., & Rodríguez-Velázquez, J. A. (2013). On the strong metric dimension of corona product graphs and join graphs. Discrete Applied Mathematics, 161(7–8), 1022–1027. https://doi.org/10.1016/j.dam.2012.10.009

Mutia, N. (2015). Dimensi Metrik Bintang dari Graf Serupa Roda. 001.

Mutianingsih, N., Asrining, U., & Uzlifah. (2016). Membandingkan Dimensi Metrik dan Dimensi Metrik Bintang. Prosiding Seminar Nasional Matematika 2016, Universita.

Saifudin, I., Umilasari, R., Program, E. S., & Jember, U. M. (2021). Automatic Aircraft Navigation Using Star Metric Dimension Theory in Fire Protected Forest Areas. 5(2), 294–304.

Saputro, S. W., Mardiana, N., & Purwasih, I. A. (2017). The metric dimension of comb product graphs. Matematicki Vesnik, 69(4), 248–258.

Yero, I. G., Kuziak, D., & Rodríguez-Velázquez, J. A. (2011). On the metric dimension of corona product graphs. Computers and Mathematics with Applications, 61(9), 2793–2798. https://doi.org/10.1016/j.camwa.2011.03.046


Refbacks

  • There are currently no refbacks.


Copyright (c) 2022 Reni Umilasari, Ilham Saifudin, Isnawati Lujeng Lestari

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