Complexity of representation of graphs by set systems

Abstract Let F be a family of subsets of S and let G be a graph with vertex set V={x A |A ∈ F } such that: ( x A , x B ) is an edge iff A⋂B≠ 0 / . The family F is called a set representation of the graph G . It is proved that the problem of finding minimum k such that G can be represented by a family of sets of cardinality at most k is NP-complete. Moreover, it is NP-complete to decide whether a graph can be represented by a family of distinct 3-element sets. The set representations of random graphs are also considered.