A circle graph is defined as the intersection graph of a set of chords of a circle.
I'm interested in any information which might help to enumerate connected circle graphs.
You can find a table enumerating the connected circle graphs up to $n=12$ at this Univ. Bergen link: Database of Circle Graphs. For example, there are 892,278,076 connected circle graphs on 12 vertices. The paper that explains how the enumeration was computed is "Interlace Polynomials: Enumeration, Unimodality, and Connections to Codes" by Lars Eirik Danielsen and Matthew G. Parker (2009); arXiv:0804.2576v2 [math.CO].