  A path in a vertex-colored graph is called \emph{vertex-rainbow} if its
internal vertices have pairwise distinct colors. A graph $G$ is \emph{rainbow
vertex-connected} if for any two distinct vertices of $G$, there is a
vertex-rainbow path connecting them. For a connected graph $G$, the
\emph{rainbow vertex-connection number} of $G$, denoted by $rvc(G)$, is defined
as the minimum number of colors that are required to make $G$ rainbow
vertex-connected. In this paper, we find all the families $\mathcal{F}$ of
connected graphs with $|\mathcal{F}|\in\{1,2\}$, for which there is a constant
$k_\mathcal{F}$ such that, for every connected $\mathcal{F}$-free graph $G$,
$rvc(G)\leq diam(G)+k_\mathcal{F}$, where $diam(G)$ is the diameter of $G$.
