The Four-Distance Domination Number in the Ladder and Star Graphs Amalgamation Result and Applications

Ilham Saifudin; Hardian Oktavianto; Lutfi Ali Muharom

doi:10.31764/jtam.v6i2.6628

The Four-Distance Domination Number in the Ladder and Star Graphs Amalgamation Result and Applications

Ilham Saifudin, Hardian Oktavianto, Lutfi Ali Muharom

Abstract

The study purpose is to determine the four-distance domination number in the amalgamation operation graph, namely the vertex amalgamation result graph of ladder graph Amal(L_m,v,n) with m≥2 and n≥2 and the vertex amalgamation result graph of a star graph with its name Amal(S_m,v,n) with m≥2 and n>2. In addition, the application use the Four-distance domination number on Jember Regency Covid-19 taskforce post-placement. The Importanceof this research, namely the optimal distribution of the Covid-19 task force post. It is not just doing mask surgeries every day on the streets. The optimal referred to can be in the form of integrated handlers in each sub-district or points that are considered to need fast handling so that coordination between posts can respond and immediately identify cases of transmission and potential infections due to interactions with patients who are already positive. The methods used in this research are pattern recognition and axiomatic deductive methods. The results of this study include:
γ_4 (Amal(S_m,v,n))=1; for m≥2 and n≥2,
γ_4 (Amal(L_m,v,n))={■(1; for 2≤m≤4 @⌊m/8⌋n+1 for m≡0,1,2,3,4 (mod 8)@⌈m/8⌉n; for others m ) ┤
and based on the Indonesia Country, Jember Regency Map, 2 Covid 19 task-force posts are needed to be placed in Balung and Kalisat sub-districts using the Four-distance domination number application.

Keywords

Domination Number; Covid-19 Task Force; Posts Distribution; Amalgamation; Operation Results Graph;

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References

Akbari Torkestani, J., & Meybodi, M. R. (2012). Finding minimum weight connected dominating set in stochastic graph based on learning automata. Information Sciences, 200, 57–77. https://doi.org/10.1016/j.ins.2012.02.057

Cerrone, C., Cerulli, R., & Golden, B. (2017). Carousel greedy: A generalized greedy algorithm with applications in optimization. Computers and Operations Research, 85(May 2018), 97–112. https://doi.org/10.1016/j.cor.2017.03.016

Citra, S. M., Kristiana, A. I., Adawiyah, R., Dafik, & Prihandini, R. M. (2021). On the packing chromatic number of vertex amalgamation of some related tree graph. Journal of Physics: Conference Series, 1836(1). https://doi.org/10.1088/1742-6596/1836/1/012025

Couturier, J.-F., Letourneur, R., & Liedloff, M. (2015). On the number of minimal dominating sets on some graph classes. Theoretical Computer Science, 562, 634–642. https://doi.org/https://doi.org/10.1016/j.tcs.2014.11.006

Enriquez, E. L. (2019). Available online at http://www.jgrma.info SUPER FAIR DOMINATING SET IN GRAPHS, (February).

Fitriani, D., & Salman, A. N. M. (2016). Rainbow connection number of amalgamation of some graphs. AKCE International Journal of Graphs and Combinatorics, 13(1), 90–99. https://doi.org/10.1016/j.akcej.2016.03.004

Gembong, A. W., Slamin, Dafik, & Agustin, I. H. (2017). Bound of Distance Domination Number of Graph and Edge Comb Product Graph. Journal of Physics: Conference Series, 855(1). https://doi.org/10.1088/1742-6596/855/1/012014

Gross, J. L., Mansour, T., & Tucker, T. W. (2014). Log-concavity of genus distributions of ring-like families of graphs. European Journal of Combinatorics, 42, 74–91. https://doi.org/https://doi.org/10.1016/j.ejc.2014.05.008

Haddadan, A., Ito, T., Mouawad, A. E., Nishimura, N., Ono, H., Suzuki, A., & Tebbal, Y. (2016). The complexity of dominating set reconfiguration. Theoretical Computer Science, 651, 37–49. https://doi.org/https://doi.org/10.1016/j.tcs.2016.08.016

Jember, T. S. C.-19. (2020). Jember Information Center. Retrieved from https://www.jember.info/dashboard-covid-19-kabupat

Jing, Y., Yang, Y., Wang, X., Song, M., & Tao, D. (2021). Amalgamating Knowledge from Heterogeneous Graph Neural Networks, 15704–15713. https://doi.org/10.1109/cvpr46437.2021.01545

Knight, A. (2020). Risk‐Assessment Frameworks. Hacking Connected Cars. https://doi.org/10.1002/9781119491774.ch8

Mohanty, J. P., Mandal, C., Reade, C., & Das, A. (2016). Construction of minimum connected dominating set in wireless sensor networks using pseudo dominating set. Ad Hoc Networks, 42, 61–73. https://doi.org/https://doi.org/10.1016/j.adhoc.2016.02.003

Nacher, J. C., & Akutsu, T. (2016). Minimum dominating set-based methods for analyzing biological networks. Methods, 102, 57–63. https://doi.org/https://doi.org/10.1016/j.ymeth.2015.12.017

Pino, T., Choudhury, S., & Al-Turjman, F. (2018). Dominating Set Algorithms for Wireless Sensor Networks Survivability. IEEE Access, 6, 17527–17532. https://doi.org/10.1109/ACCESS.2018.2819083

Poniman, B., & Fran, F. (2020). Bilangan dominasi eksentrik terhubung pada graf sunlet dan graf bishop. BIMASTER, 09(1), 71–78.

Rote, G. (2019). The maximum number of minimal dominating sets in a tree. Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms, 1201–1214. https://doi.org/10.1137/1.9781611975482.73

Umilasari, R. (2015). Bilangan Dominasi Jarak Dua Pada Graf-. ITS.

Umilasari, R., & Darmaji, D. (2017). Dominating number of distance two of corona products of graphs. Indonesian Journal of Combinatorics, 1(1), 41. https://doi.org/10.19184/ijc.2016.1.1.5

Umilasari, R., Saifudin, I., & Azhar, R. F. (2019). Optimasi Penempatan Petugas Keamanan Di Taman Safari Prigen Pasuruan Menggunakan Teori Himpunan Dominasi. JUSTINDO (Jurnal Sistem Dan Teknologi Informasi Indonesia), 4(2), 36. https://doi.org/10.32528/justindo.v4i2.2613

Unnithan, S. K. R., & Balakrishnan, K. (2019). Betweenness centrality in convex amalgamation of graphs. Journal of Algebra Combinatorics Discrete Structures and Applications, 6(1), 21–38. https://doi.org/10.13069/jacodesmath.508983

Vargas, J. G., & Kulkarni, A. (2019). Spectra of infinite graphs via freeness with amalgamation. ArXiv.

Vikade, W. D. (2016). Bilangan Dominasi Jarak Dua pada Graf Hasil Operasi. Jember: Universitas Jember.

DOI: https://doi.org/10.31764/jtam.v6i2.6628