As part of my course, I have been given an assignment that goes as follows:
"The weak dual of a plane graph $G$ is the graph obtained from the dual $G^*$ by deleting the vertex for the unbounded region of $G$. Prove that the weak dual of an outerplane graph is a forest."
While I have tried this on multiple outerplanar graphs and it holds true, I am unable to come up with a rigorous proof for the same. Any ideas on how I could proceed? I did try looking it up online but found no free/openly available source that contained the proof (although many sources mentioned it was a well known fact).