I've thought it would be interesting to have an app to play this game. The hard part would be intersection detection on beziers or splines.
Seems like the sort of thing that some clever hacker could put together in a d3 fiddle.
For example, in http://en.wikipedia.org/wiki/File:Brussel_Sprouts_Game.png, position 3, you can draw a 'diagonal' vertically through the large enclosed area, but you can also draw a graph edge between the same two vertices 'around the outside'.
Similarly, you can have disconnected parts of the graph, where it is essential to know in what 'face' they lie. For example, starting with 3 points A, B, and C, draw a circular edge from A to itself. There are two ways to do that, one where B and C end up in the same face and one where they do not.
If you store the graph as a pair (V,E), you lose that information.
[1] - https://github.com/richardTowers/Sprou.ts [2] - http://richard-towers.com/Sprou.ts/Sprouts/