At Wikipedia one reads:
"[...] most, but not all, bipartite graphs can be regarded as incidence graphs of hypergraphs.
I wonder which bipartite graphs can not be regarded as incidence graphs of hypergraphs.
The only counterexamples I can imagine are bipartite graphs in which one of the nodes representing hyperedges (node type 1) has degree less than 2 because hyperedges connect at least 2 "proper" nodes (node type 2). Proper nodes with degree less than 2 are allowed. (When loops are allowed, also hyperedge nodes with degree 1 are allowed.)
Are there other counterexamples?