Abstract. Let G be a graph. Its Wiener index W(G) is the sum of all distances in G. We show that there is a unicyclic graph on n vertices with a vertex v such that W(G)=W(G-v) if and only if n>=9. Also, there is a unicyclic graph with cycle of length c and a vertex v such that W(G)=W(G-v) if and only if c>=5. Moreover, we show that every graph H is an induced subgraph of some graph G such that W(G)=W(G-v) for some v.