On 2-rainbow domination number of functigraph and its complement

• Autorzy: Shaminezhad Athena , Vatandoost Ebrahim

Identyfikatory

DOI

10.7494/OpMath.2020.40.5.617

• Języki publikacji: EN

Abstrakty

EN

Let G be a graph and ƒ : V(G) → P({1, 2}) be a function where for every vertex v ∈ V(G), with ƒ (v) = ∅ we have [formula]. Then ƒ is a 2-rainbow dominating function or a 2RDF of G. The weight of ƒ is[formula]. The minimum weight of all 2-rainbow dominating functions is 2-rainbow domination number of G, denoted by [formula]. Let G 1 and G2 be two copies of a graph G with disjoint vertex sets V(G 1) and V(G2), and let σ be a function from V(G 1) to V(G2). We define the functigraph C(G,σ) to be the graph that has the vertex set V(C(G, ,σ)) = V(G 1) U V(G2), and the edge set [formula]. In this paper, 2-rainbow domination number of the functigraph of C(G, ,σ) and its complement are investigated. We obtain a general bound for [formula] and we show that this bound is sharp.

Słowa kluczowe

EN

2-rainbow domination number; functigraph; complement; cubic graph

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2020
• Tom: Vol. 40, no. 5
• Strony: 617--627
• Opis fizyczny: Bibliogr. 17 poz.

Twórcy

autor

Shaminezhad Athena

• Imam Khomeini International University Department of Basic Science Nouroozian Street, Qazvin, Iran

autor

Vatandoost Ebrahim

• Imam Khomeini International University Department of Basic Science Nouroozian Street, Qazvin, Iran

Bibliografia

• [1] H.A. Ahangar, J. Amjadi, S.M. Sheikholeslami, D. Kuziak, Maximal 2-rainbow domination number of a graph, AKCE Int. J. Graphs Comb. 13 (2016) 2, 157-164.
• [2] J.D. Alvarado, S. Dantas, D. Rautenbach, Averaging 2-rainbow domination and Roman domination, Discrete Appl. Math. 205 (2016), 202-207.
• [3] B. Bresar, M.A. Henning, D.F. Rall, Rainbow domination in graphs, Taiwanese J. Math. 12 (2008) 1, 213-225.
• [4] B. Bresar, T.K. Sumenjak, On the 2-rainbow domination in graphs, Discrete Appl. Math. 155 (2007) 17, 2349-2400.
• [5] G.J. Chang, J. Wu, X. Zhu, Rainbow domination on trees, Discrete Appl. Math. 158 (2010) 1, 8-12.
• [6] S. Fujita, M. Furuya, Difference between 2-rainbow domination and Roman domination in graphs, Discrete Appl. Math. 161 (2013) 6, 806-812.
• [7] M. Furuya, A note on total domination and 2-rainbow domination in graphs, Discrete Appl. Math. 184 (2015), 229-230.
• [8] F. Ramezani, E.D. Rodrguez-Bazan, J.A. Rodrguez-Velazquez, On the Roman domination number of generalized Sierpinski graphs, Filomat 31:20 (2017), 6515-6528.
• [9] Z. Shao, H. Jiang, P. Wu, S. Wang, J. Zerovnik, X. Zhang, J.B. Liu, On 2-rainbow domination of generalized Petersen graphs, Discrete Appl. Math. 257 (2019), 370-384.
• [10] Z. Stepien, M. Zwierzchowski, 2-rainbow domination number of Cartesian products: Cn□ C3 and Cn□ C5, J. Comb. Optim. 28 (2014) 4, 748-755.
• [11] T.K. Sumenjak, D.F. Rall, A. Tepeh, On k-rainbow independent domination in graphs, Appl. Math. Comput. 333 (2018), 353-361.
• [12] E. Vatandoost, F. Ramezani, Domination and signed domination number of Cayley graphs, Iran. J. Math. Sci. Inform. 14 (2019) 1, 35-42.
• [13] Y. Wang, X. Wu, N. Dehgardi, J. Amjadi, R. Khoeilar, J.B. Liu, k-rainbow domination number of P3DPn, Mathematics 7 (2019) 2, 203.
• [14] D.B. West, Introduction to Graph Theory, vol. 2, Upper Saddle River, NJ: Prentice Hall, 1996.
• [15] Y. Wu, N.J. Rad, Bounds on the 2-rainbow domination number of graphs, Graphs Combin. 29 (2013) 4, 11251133.
• [16] K.H. Wu, Y.L. Wang, C.C. Hsu, C.C. Shih, On 2-rainbow domination in generalized Petersen graphs, Int. J. Comput. Math. Comput. Syst. Theory 2 (2017) 1, 1-13.
• [17] Y. Wu, H. Xing, Note on 2-rainbow domination and Roman domination in graphs, Appl. Math. Lett. 23 (2010) 6, 706-709.

Uwagi

PL

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2020).