Bilangan Pembeda Tanpa Titik Terisolasi Graf W_n⊙K_1dan F_n⊙K_1
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Abstract
Let be a graph and be an ordered subset of the vertex set og graph The representation of a vertex in with respect to is defined as , where is the distance between vertex and for all $ The set is called a resolving set of if the representation of every vertex in is distinct. A resolving set with the minimum cardinality is called a basis of and the cardinality of a basis of is the metric dimension of the graph . A vertex in is called an isolated vertex if there are no edges incident to . A resolving set is called a non-isolated resolving set if the subgraph induced by does not contain isolated vertices. A non-isolated resolving set with the minimum cardinality is called an -basis of , and the number of its members is called the non-isolated resolving number of , nonated by .
In this paper, we discuss non-isolated resolving numbers of a graph obtained from the corona product of two graphs. The corona product of graph and graph , denoted by , is a graph obtained by taking one copy of and as many copies of as there are vertices in , then connecting every vertex from the -th copy of to the -th vertex in . The results show that if is a wheel graph or a fan graph, then the non-isolated resolving number of the corona product depends on the number of vertices in the graph
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References
[1] P.J.Slater, “Leaves of trees,” Congr. Num., vol. 14, pp. 549–559, 1975.
[2] R. A. Harary, F.; Melter, “On the metric dimension of a graph,” Ars Comb, vol. 2, pp. 191–195, 1976.
[3] S. Khuller, B. Raghavachari, and A. Rosenfeld, “Landmarks in graphs,” Discret. Appl. Math., vol. 70, no. 3, pp. 217–229, 1996, doi: 10.1016/0166-218X(95)00106-2.
[4] G. Chartrand, L. Eroh, M. A. Johnson, and O. R. Oellermann, “Resolvability in graphs and the metric dimension of a graph,” Discret. Appl. Math., vol. 105, no. 1–3, pp. 99–113, 2000, doi: 10.1016/S0166-218X(00)00198-0.
[5] E. Sebo, A; Tannier, “On Metric Generators of Graphs,” Math. Oper. Res., vol. 29, pp. 383–393, 2004.
[6] G. Chartrand and P. Zhang, “The theory and applications of resolvability in graphs: A survey,” Congr. Numer, vol. 160, pp. 47–68, 2003.
[7] M. Bača, E. T. Baskoro, A. N. M. Salman, S. W. Saputro, and D. Suprijanto, “The metric dimension of regular bipartite graphs,” Bull. Math. la Soc. des Sci. Math. Roum., vol. 54, no. 1, pp. 15–28, 2011.
[8] P. J. B. Chitra and S. Arumugam, “Resolving Sets without Isolated Vertices,” in Procedia Computer Science, 2015, vol. 74, doi: 10.1016/j.procs.2015.12.072.
[9] S. Avadayappan, M. Bhuvaneshwari, and P. J. B. Chitra, “More results on Non-isolated resolving number of a graph,” Int. J. Appl. Adv. Sci. Res. Spec. issue, pp. 49–53, 2017.
[10] S. Avadayappan, M. Bhuvaneshwari, and P. J. B. Chitra, “Non-Isolated Resolving Number for Some Splitting Graphs,” Int. J.Math. Comb., vol. 1, no. 1, pp. 9–18, 2018.
[11] W. N. Sholihah, D. Dafik, and K. Kusbudiono, “Metric Dimension dan Non-Isolated Resolving Number pada Beberapa Graf,” CGANT J. Math. Appl., vol. 2, no. 1, 2021, doi: 10.25037/cgantjma.v2i1.48.
[12] W. Abidin, “Bilangan Pembeda tanpa Titik Terisolasi Graf Cycle Books,” J. MSA (Matematika dan Stat. serta Apl., vol. 11, no. 1, pp. 118–124, 2023, doi: 10.24252/msa.v11i1.40841.
[13] I. M. Hasibuan, A. N. M. Salman, and S. W. Saputro, “Non-isolated Resolving Sets of certain Graphs Cartesian Product with a Path,” in Journal of Physics: Conference Series, 2018, vol. 1008, no. 1, doi: 10.1088/1742-6596/1008/1/012045.
[14] I. M. Hasibuan, A. N. M. Salman, and S. W. Saputro, “The Non-Isolated Resolving Number Of A Graph Cartesian Product with A Complete Graph,” Acta Math. Univ. Comenianae, vol. 91, no. 3, 2022.
[15] R. Alfarisi, D. Dafik, A. I. Kristiana, E. R. Albirri, and I. H. Agustin, “Non-isolated resolving number of graphs with homogeneous pendant edges,” in AIP Conference Proceedings, 2018, vol. 2014, doi: 10.1063/1.5054416.
[16] R. Alfarisi, Dafik, A. I. Kristiana, and I. H. Agustin, “Non-Isolated Resolving Number of Graph with Pendant Edges,” J. Interconnect. Networks, vol. 19, no. 2, 2019, doi: 10.1142/S0219265919500038.
[17] S. W. Saputro et al., “The metric dimension of the lexicographic product of graphs,” Discrete Math., vol. 313, no. 9, pp. 1045–1051, 2013, doi: 10.1016/j.disc.2013.01.021.
[18] W. Abidin, A. Salman, and S. W. Saputro, “Non-Isolated Resolving Sets of Corona Graphs with Some Regular Graphs,” Mathematics, vol. 10, no. 6, p. 962, Mar. 2022, doi: 10.3390/math10060962.
[19] W. Abidin, “Bilangan Pembeda Tanpa Titik Terisolasi Graf Hasil Operasi Korona dan Hasil Operasi Sisir Titik,” Matematika, 2023. https://digilib.itb.ac.id/gdl/view/70323#