The Šoltés problem asks for all connected graphs that keep the same Wiener index after
deletion of any graph vertex, with each deletion
leaving a connected graph (Šoltés
1991). The cycle graph 
satisfies the condition. Its Wiener
index is 165, equal to that of the path graph 
obtained by deleting any graph
vertex. The existence of another example remains open.
A Šoltés vertex of a connected graph is a graph vertex whose vertex
deletion preserves both connectedness and the Wiener
index. The Šoltés fraction is the proportion of such vertices.
Dobrynin (2026) constructed a 4-regular graph on
2400 vertices with exactly 1920 Šoltés
vertices. Its Šoltés fraction 
improves the previous 
for examples other than 
. The count uses the computed Wiener
index after each vertex deletion.
