 Theoretical Computer Science

# Recognizing line graphs of hypergraphs

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. ## Solution

I found the journal version of the preprint by Skums et al. pointed by @mhum; it is here: Discrete Mathematics 309 (2009) 35003517. There, the authors corrected their citation as follows:

The situation changes radically if one takes $k \ge 3$ instead of $k = 2$. Lovasz posed the problem of characterizing the class $L_3$, and noted that it has no characterization by a finite list of forbidden induced subgraphs (a finite characterization) . It has been proved that the recognition problems "$G \in L_k$ for $k \ge 4$ , $G \in L^l_3$ for $k \ge 3$ and the problem of recognition of edge intersection graphs of $3$-uniform hypergraphs without multiple edges  are NP-complete.

Reference 15 is the aforementioned Poljak et al. (1981).

So, I think, recognizing line graphs of $3$-hypergraphs (with multiple edges allowed) is an OPEN PROBLEM, and @mhum's answer indeed was helpful in this finding. Thanks!