Automatic Aircraft Navigation Using Star Metric Dimension Theory in Fire Protected Forest Areas
DOI:
https://doi.org/10.31764/jtam.v5i2.4331Keywords:
Forest Fires, Unique Graphs, Stars Metric Dimensions.Abstract
The purpose of this research is to determine the navigation of an unmanned aircraft automatically using theory of the metric dimension of stars in a forest fire area. The research will also be expanded by determining the star metric dimensions on other unique graphs and graphs resulting from amalgamation operations. The methods used in this research are pattern recognition and axiomatic deductive methods. The pattern detection method is to look for patterns to construct differentiated sets on the metric dimension (dim) so that the coordinate values are minimum and different. Meanwhile, axiomatic deductive is a research method that uses deductive proof principles that apply in mathematical logic by using existing axioms or theorems to solve a problem. Then the method is used to determine the stars' metric dimensions.References
Bailey, R. F., & Cameron, P. J. (2011). Base size, metric dimension and other invariants of groups and graphs. Bulletin of the London Mathematical Society, 43(2), 209–242. https://doi.org/https://doi.org/10.1112/blms/bdq096
Belmonte, R., Fomin, F. V., Golovach, P. A., & Ramanujan, M. S. (2017). Metric dimension of bounded tree-length graphs. SIAM Journal on Discrete Mathematics, 31(2), 1217–1243. https://doi.org/10.1137/16M1057383
Bollobás, B., Mitsche, D., & Prałat, P. (2013). Metric dimension for random graphs. Electronic Journal of Combinatorics, 20(4), 1–18. https://doi.org/10.37236/2639
Budianto, W. T., & Kusmayadi, T. A. (2018). The local metric dimension of starbarbell graph, Km ⊙ Pn graph, and Möbius ladder graph. Journal of Physics: Conference Series, 1008(1). https://doi.org/10.1088/1742-6596/1008/1/012050
Cáceres, J., Hernando, C., Mora, M., Pelayo, I. M., & Puertas, M. L. (2012). On the metric dimension of infinite graphs. Discrete Applied Mathematics, 160(18), 2618–2626. https://doi.org/10.1016/j.dam.2011.12.009
Epstein, L., Levin, A., & Woeginger, G. J. (2015). The (Weighted) Metric Dimension of Graphs: Hard and Easy Cases. Algorithmica, 72(4), 1130–1171. https://doi.org/10.1007/s00453-014-9896-2
Filipović, V., Kartelj, A., & Kratica, J. (2019). Edge Metric Dimension of Some Generalized Petersen Graphs. Results in Mathematics, 74(4), 182. https://doi.org/10.1007/s00025-019-1105-9
Gazol, A., Sangüesa-Barreda, G., Granda, E., & Camarero, J. J. (2017). Tracking the impact of drought on functionally different woody plants in a Mediterranean scrubland ecosystem. Plant Ecology, 218(8), 1009–1020. https://doi.org/10.1007/s11258-017-0749-3
Hakim, R. N., & Galih, B. (2019, September 23). Hampir Satu Juta Orang Menderita ISPA akibat Kebakaran Hutan dan Lahan. Www.Kompas.Com. Retrieved from https://nasional.kompas.com/read/2019/09/23/17522721/hampir-satu-juta-orang-menderita-ispa-akibat-kebakaran-hutan-dan-lahan
Halawane, J. E., Hanif, & Kinho, J. (2011). Prospek Pengembangan Jabon Merah (Anthocephalus macrophyllus) Solusi Kebutuhan Kayu Masa Depan. Manado: Balai Penelitian Kehutanan Manado.
Handa, A. K., Godinho, A., & Singh, T. (2017). Distance Antimagic Labeling of the Ladder Graph. Electronic Notes in Discrete Mathematics, 63(December), 317–322. https://doi.org/10.1016/j.endm.2017.11.028
Hauptmann, M., Schmied, R., & Viehmann, C. (2012). Approximation complexity of Metric Dimension problem. Journal of Discrete Algorithms, 14, 214–222. https://doi.org/10.1016/j.jda.2011.12.010
Jannesari, M., & Omoomi, B. (2012). The metric dimension of the lexicographic product of graphs. Discrete Mathematics, 312(22), 3349–3356. https://doi.org/10.1016/j.disc.2012.07.025
Jiang, Z., & Polyanskii, N. (2019). On the metric dimension of Cartesian powers of a graph. Journal of Combinatorial Theory. Series A, 165, 1–14. https://doi.org/10.1016/j.jcta.2019.01.002
Kelenc, A., Kuziak, D., Taranenko, A., & G. Yero, I. (2017). Mixed metric dimension of graphs. Applied Mathematics and Computation, 314(1), 429–438. https://doi.org/10.1016/j.amc.2017.07.027
Knor, M., Majstorović, S., Masa Toshi, A. T., Škrekovski, R., & Yero, I. G. (2021). Graphs with the edge metric dimension smaller than the metric dimension. Applied Mathematics and Computation, 401, 1–11. https://doi.org/10.1016/j.amc.2021.126076
Kurniawati, E. Y., Agustin, I. H., Dafik, D., Alfarisi, R., & Marsidi, M. (2018). On the local antimagic total edge chromatic number of amalgamation of graphs. AIP Conference Proceedings, 2014(September). https://doi.org/10.1063/1.5054494
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., & RodrÃguez-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
MIB. (2019). Peta dan Luas Kebakaran Hutan Lahan di Provinsi Sumsel 2019.
Mubarok, F. (2019). Kebakaran Hutan dan Lahan Terus Terjadi, Bagaimana Solusinya? Retrieved from https://www.mongabay.co.id/2019/11/16/kebakaran-hutan-dan-lahan-terus-terjadi-bagaimana-solusinya/
Mutia, N. (2015). Dimensi Metrik Bintang dari Graf Serupa Roda (Tesis). Surabaya: Tesis, Jurusan Matematika FMIPA Institut Teknologi Sepuluh Nopember (ITS).
Mutianingsih, N., Asrining, U., & Uzlifah. (2016). Membandingkan Dimensi Metrik dan Dimensi Metrik Bintang. Prosiding Seminar Nasional Matematika 2016, Universita.
Nasir, R., Zafar, S., & Zahid, Z. (2018). Ars Combinatoria ISSN : 0381-7032 Edge metric dimension of graphs, (March 2020).
Peterin, I., & Yero, I. G. (2020). Edge Metric Dimension of Some Graph Operations. Bulletin of the Malaysian Mathematical Sciences Society, 43(3), 2465–2477. https://doi.org/10.1007/s40840-019-00816-7
RodrÃguez-Velázquez, J. A., GarcÃa Gómez, C., & Barragán-RamÃrez, G. A. (2015). Computing the local metric dimension of a graph from the local metric dimension of primary subgraphs. International Journal of Computer Mathematics, 92(4), 686–693. https://doi.org/10.1080/00207160.2014.918608
RodrÃguez-Velázquez, J. A., Kuziak, D., Yero, I. G., & Sigarreta, J. M. (2015). The metric dimension of strong product graphs. Carpathian Journal of Mathematics, 31(2), 261–268.
Saputro, S. W., Suprijanto, D., Baskoro, E. T., & Salman, A. N. M. (2012). the Metric Dimension of a Graph Composition Products With Star. Journal of the Indonesian Mathematical Society, 18(2), 85–92. https://doi.org/10.22342/jims.18.2.114.85-92
Yi, E. (2013). On strong metric dimension of graphs and their complements. Acta Mathematica Sinica, English Series, 29(8), 1479–1492. https://doi.org/10.1007/s10114-013-2365-z
Downloads
Published
Issue
Section
License
Authors who publish articles in JTAM (Jurnal Teori dan Aplikasi Matematika) agree to the following terms:
- Authors retain copyright of the article and grant the journal right of first publication with the work simultaneously licensed under a CC-BY-SA or The Creative Commons Attribution–ShareAlike License.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgment of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).