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On the Roman Domination Polynomials | ||
| Communications in Combinatorics and Optimization | ||
| مقاله 15، دوره 9، شماره 3، آذر 2024، صفحه 595-605 اصل مقاله (376.35 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2023.28145.1456 | ||
| نویسندگان | ||
| Seyed Hosein Mirhoseini؛ Nader Jafari Rad* | ||
| Department of Mathematics, Shahed University, Tehran, Iran | ||
| چکیده | ||
| A Roman dominating function (RDF) on a graph $G$ is a function $ f:V(G)\to \{0,1,2\}$ satisfying the condition that every vertex $u$ with $f(u) = 0$ is adjacent to at least one vertex $v$ for which $f(v) = 2$. The weight of an RDF $f$ is the sum of the weights of the vertices under $f$. The Roman domination number, $\gamma_R(G)$ of $G$ is the minimum weight of an RDF in $G$. The Roman domination polynomial of a graph $G$ of order $n$ is the polynomial $RD(G,x)=\sum_{i=\gamma_R(G)}^{2n} d_R(G,i) x^{i}$, where $d_R(G,i)$ is the number of RDFs of $G$ with weight $i$. In this paper we prove properties of Roman domination polynomials and determine $RD(G,x)$ in several classes of graphs $G$ by new approaches. We also present bounds on the number of all Roman domination polynomials in a graph. | ||
| کلیدواژهها | ||
| Roman domination polynomial Roman dominating function؛ Roman domination number | ||
| مراجع | ||
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[1] S. Alikhani, The domination polynomial of a graph at -1, Graphs Comb. 29 (2013), no. 5, 1175–1181.
[2] S. Alikhani and N. Jafari, Some new results on the total domination polynomial of a graph, Ars Combin. 149 (2020), 185–197.
[3] S. Alikhani and Y.H. Peng, Dominating sets and domination polynomials of paths, Int. J. Math. Math. Sci. 2009 (2009), Article ID: 542040.
[4] S. Alikhani and Y.H. Peng, Introduction to domination polynomial of a graph, Ars Combin. 114 (2014), 257–266.
[5] R.A. Brualdi, Introductory Combinatorics, Pearson India, 2019.
[6] M. Chellali, N. Jafari Rad, S.M. Sheikholeslami, and L. Volkmann, Roman domination in graphs, Topics in Domination in Graphs (T.W. Haynes, S.T. Hedetniemi, and M.A. Henning, eds.), Springer, Berlin/Heidelberg, 2020, p. 365–409.
[7] M. Chellali, N. Jafari Rad, S.M. Sheikholeslami, and L. Volkmann, Varieties of Roman domination II, AKCE Int. J. Graphs Comb. 17 (2020), no. 3, 966–984.
[8] M. Chellali, N. Jafari Rad, S.M. Sheikholeslami, and L. Volkmann, Varieties of Roman domination, Structures of Domination in Graphs (T.W. Haynes, S.T. Hedetniemi, and M.A. Henning, eds.), Springer, Berlin/Heidelberg, 2021, p. 273–307.
[9] E.J. Cockayne, P.A. Dreyer Jr, S.M. Hedetniemi, and S.T. Hedetniemi, Roman domination in graphs, Discrete Math. 278 (2004), no. 1-3, 11–22.
[10] G. Deepak, A. Alwardi, and M.H. Indiramma, Roman domination polynomial of cycles, Int. J. Math. Combin. 3 (2021), 86–97.
[11] G. Deepak, A. Alwardi, and M.H. Indiramma, Roman domination polynomial of paths, Palestine J. Math. 11 (2022), no. 2, 439–448. [12] D. Gangabylaiah, M.H. Indiramma, N.D. Soner, and A. Alwardi, On the Roman domination polynomial of graphs, Bull. Int. Math. Virtual Inst. 11 (2021), no. 2, 355–365.
[13] T.W. Haynes, S.T. Hedetniemi, and P.J. Salter, Fundamentals of Domination in Graphs, Marcel Dekker, New York, 1998.
[14] D.A. Mojdeh and A.S. Emadi, Connected domination polynomial of graphs, Fasc. Math, 60 (2018), no. 1, 103–121.
[15] D.A. Mojdeh and A.S. Emadi, Hop domination polynomial of graphs, J. Discrete Math. Sci. Crypt. 23 (2020), no. 4, 825–840. | ||
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