First the author shows that the answers to two questions due to A. Neumaier are `No'. Then he shows how to modifiy the second question in order to obtain an affirmative answer. In fact he proves (in design theoretical terminology) the following: Let \(G\) be some (connected) graph where each neighborhood induces a line graph and where each edge is contained in some triangle. Then \(G\) is the underlying graph of some (connected) hypergraph (i.e. set system) where any two distinct vertices lie in 0 or 2 common hyperedges and any two distinct hyperedges contain at most 2 common vertices.