  A path in a vertex-colored graph $G$ is \emph{vertex rainbow} if all of its
internal vertices have a distinct color. The graph $G$ is said to be
\emph{rainbow vertex connected} if there is a vertex rainbow path between every
pair of its vertices. Similarly, the graph $G$ is \emph{strongly rainbow vertex
connected} if there is a shortest path which is vertex rainbow between every
pair of its vertices. We consider the complexity of deciding if a given
vertex-colored graph is rainbow or strongly rainbow vertex connected. We call
these problems \probRvc and \probSrvc, respectively. We prove both problems
remain NP-complete on very restricted graph classes including bipartite planar
graphs of maximum degree 3, interval graphs, and $k$-regular graphs for $k \geq
3$. We settle precisely the complexity of both problems from the viewpoint of
two width parameters: pathwidth and tree-depth. More precisely, we show both
problems remain NP-complete for bounded pathwidth graphs, while being
fixed-parameter tractable parameterized by tree-depth. Moreover, we show both
problems are solvable in polynomial time for block graphs, while \probSrvc is
tractable for cactus graphs and split graphs.
