On the preservation of the Wiener index of cubic graphs upon vertex removal

The Wiener index, $W(G)$, is the sum of distances between all vertices of a connected graph $G$. In 2018, Majstorović, Knor and Škrekovski posed the problem of finding $r$-regular graphs except cycle $C_{11}$ having at least one vertex $v$ with property $W(G)=W(G-v)$. An infinite family of cubic graphs with four such vertices is constructed.