An intersection graph is a graph obtained from an indexed family 
of subsets of a set 
. It has vertices 
, with 
and 
adjacent whenever 
and 
. The representing subsets need
not be distinct and may be empty.
Every graph has such a set-intersection representation. The graph intersection number is the
minimum possible size of 
. Requiring the representing subsets to be distinct gives a
different variant of the representation problem (Erdős, Goodman, and Pósa
1966).
