Set Labelling Vertices To Ensure Adjacency Coincides With Disjointness

Given a set of nonempty subsets of some universal set, their intersection graph is defined as the graph with one vertex for each set and two vertices are adjacent precisely when their representing sets have non-empty intersection. Sometimes these sets are finite, but in many well known examples like geometric graphs (including interval graphs) they are infinite. One can also study the reverse problem of expressing the vertices of a given graph as distinct sets in such a way that adjacency coincides with intersection of the corresponding sets. The sets are usually required to conform to some template, depending on the problem, to be either a finite set, or some geometric set like intervals, circles, discs, cubes etc. The problem of representing a graph as an intersection graph of sets was first introduced by Erdos [Alon, Noga, Covering graphs by the minimum number of equivalence relations, Combinatorica 6(1986), 201-206] and they looked at minimising the underlying universal set necessary to represent any given graph. In that paper it was shown that the problem is NP complete. In this paper we study a natural variant of this problem which is to consider graphs where vertices represent distinct sets and adjacency coincides with disjointness. Although this is nearly the same problem on the complement graph, for specific families of graphs this is a more natural way of viewing it. The parameter we take into account is the minimum universe size possible (disregarding individual label sizes).