  The Wiener index, $W(G)$, of a connected graph $G$ is the sum of distances
between its vertices. In 2021, Akhmejanova et al. posed the problem of finding
graphs $G$ with large $R_m(G)= |\{v\in V(G)\,|\,W(G)-W(G-v)=m \in \mathbb{Z}
\}|/ |V(G)|$. It is shown that there is a graph $G$ with $R_m(G) > 1/2$ for any
integer $m \ge 0$. In particular, there is a regular graph of even degree with
this property for any odd $m \ge 1$. The proposed approach allows to construct
new families of graphs $G$ with $R_0(G) \rightarrow 1/2$ when the order of $G$
increases.
