Abstract: A vertex-colored graph G is said to be rainbow vertex-connected if every two vertices of G are connected by a path whose internal vertices have distinct colors, such a path is called a rainbow path. The rainbow vertex-connection number of a connected graph G, denoted by rvc（G）, is the smallest number of colors that are needed in order to make G rainbow vertex-connected. If for every pair u, v of distinct vertices, G contains a rainbow u-v geodesic, then G is strong rainbow vertex-connected. The minimum number k for which there exists a k-vertex-coloring of G that results in a strongly rainbow vertex-connected graph is called the strong rainbow vertex-connection number of G, denoted by srvc（G）. Observe that rvc（G） ≤ srvc（G） for any nontrivial connected graph G. In this paper, for a Ladder Ln,we determine the exact value of srvc（L_n） for n even. For n odd, upper and lower bounds of srvc（L_n） are obtained. We also give upper and lower bounds of the（strong） rainbow vertex-connection number of Mbius Ladder. 

Key words: vertex-coloring, rainbow vertex-connection, (strong) rainbow vertex-connection number, Ladder, M?obius Ladder