A planar graph is called convex, if it can be drawn in a way such that every face, including the outer face is convex.
Wikipedia states that a planar graph is convex if and only if it is a subdivision of a $3$-connected planar graph. What about a cycle ? It is not a subdivision of a $3$-connected planar graph, but is convex.
- Is it true that the criterion does not hold for cycles, or do I something wrong ?
- Are the cycles the only exceptions ?
I don't know what theorem you are talking about, but from what you have written it seems that wikipedia is right. Cycles according to your definition are not convex planar graphs, because if you draw a cycle in a way that the interior face is convex, then the outer face will not be convex, and vice versa. See the attached image, the interior face is convex, but the outer face is not because the ends of the given segment belong to the outer face, but the whole segment does not.