Unicyclic graphs with strong equality between the 2-rainbow domination and independent 2-rainbow domination numbers
نویسندگان
1 Azarbaijan Shahid Madani University
2 Azarbaijan Shahid Madani University
3 University of Blida LAMDA-RO Laboratory
4 Azarbaijan Shahid Madani University
doi
10.22108/toc.2015.6518چکیده
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 $\gamma_{r2}(G)\equiv i_{r2}(G)$.
