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A vertex-colored graph $G$ is {\it rainbow vertex-connected} if any pair ofvertices in $G$ are connected by a path whose internal vertices have distinctcolors, which was introduced by Krivelevich and Yuster. The {\it rainbowvertex-connection number} of a connected graph $G$, denoted by $rvc(G)$, is thesmallest number of colors that are needed in order to make $G$ rainbowvertex-connected. In a previous paper we showed that it is NP-Complete todecide whether a given graph $G$ has $rvc(G)=2$. In this paper we show that forevery integer $k\geq 2$, deciding whether $rvc(G)\leq k$ is NP-Hard. We alsoshow that for any fixed integer $k\geq 2$, this problem belongs to NP-class,and so it becomes NP-Complete.

Further hardness results on the rainbow vertex-connection number of graphs