Let $G=(V,E)$ be a graph with the vertex set $V=V(G)$ and the edge set $E=E(G)$. Let $k$ be a positive integer and $\gamma_{rk}(G)$ ($\gamma_{i_{rk}}(G)$) be $k$-rainbow domination (independent $k$-rainbow domination) number of a graph $G$. In this paper, we study the $k$-rainbow domination and independent $k$-rainbow domination numbers of graphs. We obtain bounds for $\gamma_{rk}(G-e)$ ($\gamma_{i_{rk}}(G-e)$) in terms of $\gamma_{rk}(G)$ ($\gamma_{i_{rk}}(G)$). Finally, the relation between weak $3$-domination and $3$-rainbow domination number of graphs will be investigated.
