Suppose e is contained in a circle ci, then this edge is deleted, but the two adjacent points associated with this edge are still connected, so the connectivity of the original image is not destroyed. So it's not trimming and contradiction. So the assumption is not true, that is, e is not included in any circle of g; ?
2)e is contained in any circle of G? = > e is trimming?
Assuming that e is not a cut edge, the generated subgraph is still connected by deleting this edge. There is a basic road between two adjacent points associated with e, which forms a circle with e. Contradict with the topic. So the assumption is not true, that is, e is trimming.
According to 1) and 2), the conclusion holds.