Metric Dimension of Graph Join P2 and Pt

Loeky Haryanto, Nurdin Nurdin, Hasmawati Hasmawati

Abstract


The following metric dimension of join two paths $P_2 + P_t$ is determined as follows. For every $k = 1, 2, 3, ...$ and $t = 2 + 5k$ or $t = 3 + 5k$, the dimension of $P_2 + P_t$ is $2 + 2k$ whereas for $t = 4 + 5k, t = 5(k+1)$ or $t = 1 + 5(k+1)$, the dimension is $3 + 2k$. In case $t \geq 7$, the dimension is determined by a chosen (maximal) ordered basis for $P_2 + P_t$ in which the integers 1, 2 are the two consecutive vertices of $P_2$ and the next integers $3, 4, ..., t + 2$ are the $t$ consecutive vertices of $P_t$. If $t \geq 10$, the ordered binary string contains repeated substrings of length 5. For $t < 7$, the dimension is easily found using a computer search, or even just using hand computations.

Full Text:

PDF

References


bibitem{Chartrand} Chartrand, G., Eroh, L., Johnson, M. A., and Oellermann, O. R. , “Resolvability in graphs and the metric dimension of a graph”, {em Discrete Appl. Math}, textbf{105} (2000), 99-113.

bibitem{Gallian} Gallian, J. A. “A Dynamic Survey of Graph Labeling”, {em Electron J Comb.}, 19-th Ed. (2016), 1-408.

bibitem{Harary} Harary, F. and Melter, R. A. , “On the metric dimension of a graph”, {em Ars Combinatoria} textbf{2} (1976), 191 - 195.

bibitem{Kuziak} Kuziak, D., Rodrígues-Veláquez, J.A., and Yero, I. G.,“Computing the metric dimension of a graph from primary subgraphs”, {em Discuss. Math.} textbf{37} (Issue 1, 2017), 273 - 293

bibitem{KuziakB} Kuziak, D., Yero, I.G., and Rodrígues-Veláquez, J.A., "On the strong metric dimension of corona product graphs and join graphs", Discrete Appl. Math, textbf{161} (Issue 7, 2013), Pages 1022-1027

bibitem{Slater} Slater, P. J., “Leaves of trees”, {em Congr. Numer.} textbf{14} (1975), 549 - 559

end{thebibliography}




DOI: https://doi.org/10.22342/jims.25.1.747.75-84

Refbacks

  • There are currently no refbacks.



Journal of the Indonesian Mathematical Society
Mathematics Department, Universitas Gadjah Mada
Senolowo, Sinduadi, Mlati, Sleman Regency, Special Region of Yogyakarta 55281, Telp. (0274) 552243
Email: jims.indoms@gmail.com


p-ISSN: 2086-8952 | e-ISSN: 2460-0245


Journal of the Indonesian Mathematical Society is licensed under a Creative Commons Attribution 4.0 International License

web statistics
View My Stats