It turns out that the restriction to intersection graphs of Helly hypergraphs is the most important reason for making things (first of all, recognition) easier. We may not know all star graphs, but many of them.

Now, many natural properties **P** have the property
that every system of **P**-sets has the Helly-property.
Examples are intervals of the real line, or more generally,
*d*-dimensional boxes
Also, subtrees of a given tree fulfill the Helly-property.
This property is named after the Austrian mathematician Eduard Helly.
Helly showed in 1923 that whenever every *d+1* sets of a collection
of convex sets in **R ^{d}** have nonempty intersection,
then the total intersection of these sets must be nonempty too.
That coincides with our definition only for

The dual notion of `Helly' is `conformal'.
A hypergraph *H* is called **conformal** if its dual *H**
has the Helly-property.
Another formulation is: Every clique of the underlying graph of *H*
should be covered by some hyperedge, that is,
The clique hypergraph
of the underlying graph of *H* should be contained in *H*.
Now it is very easy to convince you that every graph is not only an
intersection graph, but even an intersection graph of some Helly-hypergraph,
since every graph is the underlying graph of its own (conformal)
clique hypergraph.

See examples how to recognize
interval graphs
respectively line graphs with this approach.
However, the approach
fails for intersection graphs
of boxes in **R ^{2}**.

In models with Helly property, the topology of the union of all sets considered is related to the topology of the intersection graph. Or go back to the start page for intersection graphs.

Erich Prisner

made on January 12, 1999