Consider a circular drawing of a simple (in particular, loopless) graph $G$ in which edges are drawn as straight lines inside the circle. The crossing graph for such a drawing is the simple graph whose nodes correspond to the edges of $G$ and in which two nodes are adjacent if and only if the corresponding edges cross.
Example. The graph $G$ has four vertices (1–4) and three edges (a–c) where $a = 12$, $b = 13$, $c = 24$. In the circular drawing, $b$ and $c$ cross, so the crossing graph has three nodes and a single edge $bc$.
Here are my questions:
Is every simple graph the crossing graph of some circular graph drawing?
If not, how does a counterexample look like?
If yes, how can such a graph drawing be constructed?