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On the Roman domination number of the subdivision of some graphs | ||
| Journal of Discrete Mathematics and Its Applications | ||
| دوره 9، شماره 4، اسفند 2024، صفحه 321-333 اصل مقاله (295.77 K) | ||
| نوع مقاله: Full Length Article | ||
| شناسه دیجیتال (DOI): 10.22061/jdma.2024.11309.1099 | ||
| نویسندگان | ||
| Rostam Yarke Salkhori* ؛ Ebrahim Vatandoost؛ Ali Behtoei | ||
| Imam Khomeini International University, P.O. Box 34148- 96818, Qazvin, I. R. Iran | ||
| تاریخ دریافت: 05 آبان 1403، تاریخ بازنگری: 30 آبان 1403، تاریخ پذیرش: 11 آذر 1403 | ||
| چکیده | ||
| A Roman dominating function on a graph $G = (V, E)$ is a function $f : V(G) → {0, 1, 2}$ satisfying the condition that every vertex u for which $f(u) = 0$ is adjacent to at least one vertex v for which $f(v) = 2$. The weight of a Roman dominating function is the value $f(V) = \sum_{u∈V(G)}f(u)$. The minimum possible weight of a Roman dominating function on $G$ is called the Roman domination number of $G$ and is denoted by $\gamma_R(G)$. In this paper, and among some other results, we provide some bounds for the Roman domination number of the subdivision graph $S(G)$ of an arbitrary graph $G$. Also, we determine the exact value of $\gamma_R(S(G))$ when $G$ is $K_n$, $K_{r,s} or $K_{n_1,n_2,...,n_k}$. | ||
| کلیدواژهها | ||
| Roman dominating function؛ bipartite graph؛ tree | ||
| مراجع | ||
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[1] S. Ahmadi, E. Vatandoost, A. Behtoei, Domination number and identifying code number of the subdivision graphs, J. Algebra. Syst. Articles in Press. https://doi.org/10.22044/jas.2023.12257.1649 [2] W. Barrett, S. Butler, M. Catral, S. M. Fallat, H. T. Hallk, L. Hogben, M. Young, The maximum nullity of a complete subdivision graph is equal to its zero forcing number, Electron. J. Linear Algebra 27 (2014) 444–457. https://doi.org/10.13001/1081-3810.1629 [3] M. Chellali, N. Jafari Rad, S. M. Sheikholeslami, L. Volkmann, Roman domination in graphs, In: Topics in Domination in Graphs, Eds. T.W. Haynes, S.T. Hedetniemi and M.A. Henning, Springer 2020 365–409.https://doi.org/10.1007/978-3-030-51117-3 11 [4] M. Chellali, N. Jafari Rad, S. M. Sheikholeslami, L. Volkmann, Varieties of Roman domination, In: Structures of Domination in Graphs, Eds. T.W. Haynes, S.T. Hedetniemi and M.A. Henning, Springer 2021, 273-307. https://doi.org/10.1007/978-3-030-58892-2 10 [5] M. Chellali, N. Jafari Rad, S. M. Sheikholeslami, L. Volkmann, Varieties of Roman domination II, AKCE Int. J. Graphs Comb. 17 (2020) 966–984.https://doi.org/10.1016/j.akcej.2019.12.001 [6] M. Chellali, N. Jafari Rad, S. M. Sheikholeslami, L. Volkmann, The Roman domatic problem in graphs and digraphs: A survey, Discuss. Math. Graph Theory 42 (2022) 861–891. https://doi.org/10.7151/dmgt.2313 [7] E. J. Cockayne, P. M. Dreyer, S. M. Hedetniemi, S. T. Hedetniemi, Roman domination in graphs, Discrete Math. 278 (2004) 11–22.https://doi.org/10.1016/j.disc.2003.06.004 [8] S. M. Mirafzal, Some algebraic properties of the subdivision graph of a graph, Commun. Comb. Optim. 9 (2024) 297–307. https://doi.org/10.22049/cco.2023.28270.1494 [9] R. Y. Salkhori, E. Vatandoost, A. Behtoei, 2-Rainbow domination number of the subdivision of graphs, Commun. Comb. Optim. Articles in Press https://doi.org/10.22049/cco.2024.28850.1749 [10] Y. Wu, N. J. Rad, Bounds on the 2−rainbow domination of graph, Graphs Combin. 29 (2013) 25–33. https://doi.org/10.1007/s00373-012-1158-y [11] T. Zec, On the Roman domination problem of some Johnson graphs, Filomat 37 (2023) 2067–2075. https://doi.org/10.2298/fil2307067z | ||
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