The dual graph of a plane graph G is a graph that has a vertex for each face of G. The dual graph has an edge for each pair of faces in G that are separated from each other by an edge, and a self-loop when the same face appears on both sides of an edge.
I was going to do exercise 6.1.18 when I was reading West's graph theory textbook.
I noticed (c), and I think I found a counterexample. I don't know if that's right.
I found a graph $G$ with a cut edge.
First I figured out his dual graph $G^*$ as shown below in blue.
When I go further and try to find the dual graph of $G^*$, I find that there is another graph that is not isomorphic with $G$.
I don't know what's wrong with my thinking. I hope you can give me some advice. Thanks！
Note that as you've drawn it, the dual you've constructed would even be a counterexample to part (b). I haven't checked in detail whether part (c) still works out here, but I think you're intended to only cross the bridge once with the dual edges -- pardon the MSPaint diagram: