This is equivalent to the two statements above. Look at our example on how the star graphs of an intersection graph may look like. Intersection graphs can also be defined by duality. Note that in geometric intersection models there may be infinitely many star graphs, however only finitely many distinct. Moreoever, since most of these geometric models are stable under slight pertuberations, we may assume in most cases that every star graph occuring occurs infinitely often.
Duality is used for some natural approach for the recognition problem.
Now you may be interested in the special role of cliques in intersection graphs or go back to the start page for intersection graphs.