Introduction:
The line graph of a hypergraph is the graph whose vertex set is the set of the hyperedges of the hypergraph $\{E_1,...E_m\}$, with two hyperedges adjacent when they have
a nonempty intersection.
-$L(H) \subset G$ :: any graph is the line graph of some hypergraph, but the opposite case is NOT always true (that is, not any hypergraph is the line graph of some graph).
Example: $K_{1,3}$ is the line graph of the hypergraph $(\{v_1,v_2,v_3\},$ $\{\{v_1,v_2,v_3\}, \{v_1\}, \{v_2\}, \{v_3\}\})$ but by Beineke's characterization of line graphs it is not the line graph of any graph.
Question?
Is there is any condition on the hypergraph $H$ (even in very special cases) such that $L(H)=L(G),$ that is the line graph of the hypergraph $H$ is also a line graph of some multigraph $G$.
Any idea will be useful!