图$G$的$2$-彩虹控制函数定义为从$G$的顶点集$V(G)$到集合$\{1,2\}$的幂集的函数$f$使得对任意满足$f(v)=\varnothing$的顶点$v$, 均有$\bigcup_{u\in N(v)}f(u)=\{1,2\}$成立, 其中$N(v)$是顶点$v$的邻域。称$\sum_{v\in V(G)}|f(v)|$是图$G$的$2$-彩虹控制函数$f$的权。图$G$的$2$-彩虹控制数是指$G$的$2$-彩虹控制函数的最小权。通过对图的结构分析, 利用图的顶点数、周长、围长以及最小度得到了图的$2$-彩虹控制数的一些新的上下界。

A $2$-rainbow dominating function of a graph $G$ is a function $f$ from the vertex set $V(G) $ of $G $ to the power set of the set $\{1,2\} $ such that any vertex $v$ with $f(v)=\varnothing$ satisfies that $\bigcup_{u\in N(v)}f(u)=\{1,2\} $, where $N(v) $ is the neighborhood of $v$. The value $\sum_{v\in V(G)}|f(v)| $ is called the weight of a $2$-rainbow dominating function $f $ of $G$. The $2$-rainbow domination number of $G $ is the minimum weight of a $2$-rainbow dominating function of $G$. By the structure analysis of graphs, some new lower and upper bounds on 2-rainbow domination number of graphs are derived in terms of the number of vertices, circumference, girth and minimum degree.