The line graph of a hypergraph $H$ is the (simple) graph $G$ having edges of $H$ as vertices with two edges of $H$ are adjacent in $G$ if they have nonempty intersection. A hypergraph is an $r$-hypergraph if each of its edges has at most $r$ vertices.
What is the complexity of the following problem: Given a graph $G$, does there exist a $3$-hypergraph $H$ such that $G$ is the line graph of $H$?
It is well-known that recognizing line graphs of $2$-hypergraph is polynomial, and it is known (by Poljak et al., Discrete Appl. Math. 3(1981)301-312) that recognizing line graphs of $r$-hypergraphs is NP-complete for any fixed $r \ge 4$.
Note: In case of simple hypergraphs, i.e. all hyperedges are distinct, the problem is NP-complete as proved in the paper by Poljak et al.