  The \emph{Wiener index} is one of the most widely studied parameters in
chemical graph theory. It is defined as the sum of the lengths of the shortest
paths between all unordered pairs of vertices in a given graph. In 1991,
\v{S}olt\'es posed the following problem regarding the Wiener index: Find all
graphs such that its Wiener index is preserved upon removal of any vertex. The
problem is far from being solved and to this day, only one graph with such
property is known: the cycle graph on 11 vertices.
  In this paper, we solve a relaxed version of the problem, proposed by Knor et
al.\ in 2018. For a given $k$, the problem is to find (infinitely many) graphs
having exactly $k$ vertices such that the Wiener index remains the same after
removing any of them. We call these vertices \emph{good} vertices and we show
that there are infinitely many cactus graphs with exactly $k$ cycles of length
at least 7 that contain exactly $2k$ good vertices and infinitely many cactus
graphs with exactly $k$ cycles of length $c \in \{5,6\}$ that contain exactly
$k$ good vertices. On the other hand, we prove that $G$ has no good vertex if
the length of the longest cycle in $G$ is at most $4$.
