A $2$-\emph{rainbow dominating function} (2RDF) on a graph $G=(V,E)$ is a function $f$ from the vertex set $V$ to the set of all subsets of the set $\{1,2\}$ such that for any vertex $v\in V$ with $f(v)=\emptyset$ the condition $\bigcup_{u\in N(v)}f(u)=\{1,2\}$ is fulfilled. A 2RDF $f$ is independent (I2RDF) if no two vertices assigned nonempty sets are adjacent. The weight of a 2RDF $f$ is the value $\omega(f)=\sum_{v\in V}|f (v)|$. The 2-\emph{rainbow domination number} $\gamma_{r2}(G)$ (respectively, the independent $2$-rainbow domination number $i_{r2}(G)$) is the minimum weight of a 2RDF (respectively, I2RDF) on $G$. We say that $\gamma_{r2}(G)$ is strongly equal to $i_{r2}(G)$ and denote by $\gamma_{r2}(G)\equiv i_{r2}(G)$, if every 2RDF on $G$ of minimum weight is an I2RDF. In this paper we characterize all unicyclic graphs $G$ with $\gamma_{r2}(G)\equiv i_{r2}(G)$.
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Amjadi, J. , Chellali, M. , Falahat, M. and Sheikholeslami, S. M. (2015). Unicyclic graphs with strong equality between the 2-rainbow domination and independent 2-rainbow domination numbers. Transactions on Combinatorics, 4(2), 1-11. doi: 10.22108/toc.2015.6518
MLA
Amjadi, J. , , Chellali, M. , , Falahat, M. , and Sheikholeslami, S. M. . "Unicyclic graphs with strong equality between the 2-rainbow domination and independent 2-rainbow domination numbers", Transactions on Combinatorics, 4, 2, 2015, 1-11. doi: 10.22108/toc.2015.6518
HARVARD
Amjadi, J., Chellali, M., Falahat, M., Sheikholeslami, S. M. (2015). 'Unicyclic graphs with strong equality between the 2-rainbow domination and independent 2-rainbow domination numbers', Transactions on Combinatorics, 4(2), pp. 1-11. doi: 10.22108/toc.2015.6518
CHICAGO
J. Amjadi , M. Chellali , M. Falahat and S. M. Sheikholeslami, "Unicyclic graphs with strong equality between the 2-rainbow domination and independent 2-rainbow domination numbers," Transactions on Combinatorics, 4 2 (2015): 1-11, doi: 10.22108/toc.2015.6518
VANCOUVER
Amjadi, J., Chellali, M., Falahat, M., Sheikholeslami, S. M. Unicyclic graphs with strong equality between the 2-rainbow domination and independent 2-rainbow domination numbers. Transactions on Combinatorics, 2015; 4(2): 1-11. doi: 10.22108/toc.2015.6518
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