Abstract
Let \(G=(V,E)\) be a simple graph with no isolated vertex. A 2-rainbow dominating function (2RDF) of G is a function f from the vertex set V(G) to the set of all subsets of the set \(\{1,2\}\) such that for any vertex \(v\in V(G)\) with \(f(v)=\emptyset\) the condition \(\bigcup _{u\in N(v)}f(u)=\{1,2\}\) is fulfilled, where N(v) is the open neighborhood of v. A 2-rainbow dominating function f is called a total 2-rainbow dominating function (T2RDF) if the subgraph of G induced by \(\{v \in V(G) \mid f (v) \not =\emptyset \}\) has no isolated vertex. The weight of a T2RDF f is defined as \(w(f)= \sum _{v\in V(G)} |f(v)|\). The total 2-rainbow domination number, \(\gamma _{tr2}(G)\), is the minimum weight of a total 2-rainbow dominating function on G. In this paper, we characterize all graphs G whose total 2-rainbow domination number is equal to their order minus one.
Similar content being viewed by others
Change history
27 February 2018
The original version of this article unfortunately contained misprint.
References
Abdollahzadeh Ahangar H, Amjadi J, Sheikholeslami SM, Soroudi M (2017) On the total Roman domination number of graph. Ars Combin (to appear)
Abdollahzadeh Ahangar H, Henning MA, Samodivkin V, Yero IG (2016) Total Roman domination in graphs. Appl Anal Discret Math 10:501–517
Abdollahzadeh Ahangar H, Amjadi J, Jafari Rad N, Samodivkin V (2018a) Total k-rainbow domination numbers in graphs. Commun Comb Optim 3:37–50
Abdollahzadeh Ahangar H, Amjadi J, Chellali M, Nazari-Moghaddam S, Shahbazi L, Sheikholeslami SM (2018b) Total 2-rainbow domination numbers in trees (submitted)
Chang GJ, Wu J, Zhu X (2010) Rainbow domination on trees. Discret Appl Math 158:8–12
Chellali M, Jafari Rad N (2013) On \(2\)-rainbow domination and Roman domination in graphs. Australas J Comb 56:85–93
Chunling T, Xiaohui L, Yuansheng Y, Meiqin L (2009) 2-rainbow domination of generalized Petersen graphs \(P(n; 2)\). Discret Appl Math 157:1932–1937
Brešar B, Henning MA, Rall DF (2008) Rainbow domination in graphs. Taiwan J Math 12:201–213
Brešar B, Sumenjak TK (2007) On the 2-rainbow domination in graphs. Discret Appl Math 155:2394–2400
Dehgardi N, Sheikholeslami SM, Volkmann L (2015) The rainbow domination subdivision numbers of graphs. Mat Vesnik 67:102–114
Falahat M, Sheikholeslami SM, Volkmann L (2014) New bounds on the rainbow domination subdivision number. Filomat 28:615–622
Haynes TW, Hedetniemi ST, Slater PJ (1998) Fundamentals of domination in graphs. Marcel Dekker, New York
Haynes TW, Hedetniemi ST, Slater PJ (1998) Domination in graphs: advanced topics. Marcel Dekker Inc., New York
Henning MA, Yeo A (2013) Total domination in graphs. New York, Springer
Liu C-H, Chang GJ (2013) Roman domination on strongly chordal graphs. J Comb Optim 26:608–619
Meierling D, Sheikholeslami SM, Volkmann L (2011) Nordhaus–Gaddum bounds on the \(k\)- rainbow domatic number of a graph. Appl Math Lett 24:1758–1761
Sheikholeslami SM, Volkmann L (2012) The \(k\) -rainbow domatic number of a graph. Discuss Math Graph Theory 32:129–140
Wu Y, Jafari Rad N (2013) Bounds on the 2-rainbow domination number of graphs. Graphs Comb 29:1125–1133
Wu Y, Xing H (2010) Note on 2-rainbow domination and Roman domination in graphs. Appl Math Lett 23:706–709
Xu G (2009) 2-rainbow domination of generalized Petersen graphs \(P(n; 3)\). Discret Appl Math 157:2570–2573
Acknowledgements
The authors are grateful to anonymous referees for their remarks and suggestions that helped improve the manuscript.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ahangar, H.A., Khaibari, M., Rad, N.J. et al. Graphs with Large Total 2-Rainbow Domination Number. Iran J Sci Technol Trans Sci 42, 841–846 (2018). https://doi.org/10.1007/s40995-017-0465-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40995-017-0465-9
Keywords
- 2-rainbow dominating function
- 2-rainbow domination number
- Total 2-rainbow dominating function
- Total 2-rainbow domination number