I was, and still am, out seeking a way to get the adjacency matrix of the dual of $G$ without pen and paper. hardmath answered that the way I posted the question, it doesn't have a unique dual graph and it's therefore not possible to construct the adjacency matrix. But he also gave me useful information about how to fix the question to get an answer to my question.
Now I have two options:
- Edit the question, but this would make hardmath's answer look a little strange.
- Repost another question with the additional $3$-connectedness condition, which would be close to a duplicate to the one linked above.
So, both don't perfectly fit. I found this "edit and rephrase or ask new question?" on meta, but the answers don't fit to my case. What to do?