# Representing non-planar graphs with overlapping circles

We know that we can represent any planar graph by a set of circles in the plane, known as a coin graph. Each circle represents a vertex and there is an edge between two vertices if and only if the circles "kiss" at their boundary.

Suppose that instead we allow the circles to overlap, and represent an edge by a pair of circles that intersect in their interior? What class of graphs can we represent in this model? Clearly we can represent complete graphs (every circle intersects every other circle). Can we represent all graphs like this?

The definitive article is a paper by Hlineny and Kratochvil from 2001. In it they show that the problem of recognizing a disk intersection graph (your question) is NP-hard, which suggests that it will be difficult to come up with a clean characterization. They also point out that $K_{3,3}$ cannot be represented as the intersection of disks, answering the other part of your question.
In a paper with McDiarmid we showed that the number of labelled graphs on $n$ vertices that are intersection graphs of disks is $n^{3n} \cdot \Theta(1)^n$ which is far less than $2^{{n\choose 2}}$, the total number of labelled graphs on $n$ vertices, and much more that $n^n \Theta(1)^n$, the number of planar graphs (touching graphs of disks) on $n$ vertices.
(Here I mean by $\Theta(1)^n$ a quantity that is bounded below by $c^n$ and above by $C^n$ for some constants $c,C > 0$.)