Abstract. We study graphs G having a vertex v such that W(G)=W(G-v), where W is the Wiener index. In an earlier paper we found that there are infinitely many such graphs with vertex v of degree 2. In this paper we show that for every k>=2 there are infinitely many graphs G with a vertex v of degree k satisfying the above property. Analogous statements hold if k is n-1 or n-2, where n is the number of vertices of G. We also show that dense graphs do not satisfy the above relation.