Šoltés' hypergraphs

More than 30 years ago, Šoltés observed that the total distance of the graph C₁₁ does not change by deleting a vertex, and wondered about the existence of other such graphs, called Šoltés graphs. We extend the definition of Šoltés’ graphs to Šoltés’ hypergraphs, determine all orders for which a Šoltés’ hypergraph exists, construct infinitely many uniform Šoltés’ hypergraphs, and identify the one with minimum size.