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For a vertex set $S\\subseteq V(G)$, a tree that connects $S$ in $G$ is called an {\\it $S$-tree}. The minimum number of colors that are needed in an edge-coloring of $G$ such that there is a rainbow $S$-tree for every $k$-set $S$ of $V(G)$ is called the {\\it $k$-rainbow index} of $G$, denoted by $rx_k(G)$. 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