A relaxed version of Šoltés's problem and cactus graphs

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 $4$.