  The Wiener index $W(G)$ of a connected graph $G$ is a sum of distances
between all pairs of vertices of $G$. In 1991, \v{S}olt\'{e}s formulated the
problem of finding all graphs $G$ such that for every vertex $v$ the equation
$W(G)=W(G-v)$ holds. The cycle $C_{11}$ is the only known graph with this
property. In this paper we consider the following relaxation of the original
problem: find a graph with a large proportion of vertices such that removing
any one of them does not change the Wiener index of a graph. As the main
result, we build an infinite series of graphs with the proportion of such
vertices tending to $\frac{1}{2}$.
