We have seen that intersection graphs of Helly-hypergraphs behave well. These hypergraphs are just the clique hypergraphs of graphs.
G is the intersection graph of the family of all cliques of G .
Thus the intersection graphs of conformal hypergraphs are just the clique graphs. Clique graphs seem to be very difficult to deal with. In other words, clique graphs are very interesting. Let G be the intersection graph of certain P -sets S x . Let us give the criterion for clique graphs of intersection graphs of 2-dimensional boxes.