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The Wiener index $W(G)$ of a connected graph $G$ is a sum of distancesbetween all pairs of vertices of $G$. In 1991, \v{S}olt\'{e}s formulated theproblem 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 thisproperty. In this paper we consider the following relaxation of the originalproblem: find a graph with a large proportion of vertices such that removingany one of them does not change the Wiener index of a graph. As the mainresult, we build an infinite series of graphs with the proportion of suchvertices tending to $\frac{1}{2}$.

Wiener index and graphs, almost half of whose vertices satisfy \v{S}olt\'{e}s property