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. The question is, whether the vertices of the graph H in question can be placed on the plane in such a way that for every two adjacent vertices x and y , the box generated by them covers only vertices of H adjacent to both x and y . Thus we place the vertices of H in such a way that the box generated by the placed vertices of every clique of H contains no further placed vertices of H .