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).
