Wikipedia claims (permanent link) without reference:
Testing whether one planar graph is dual to another is NP-complete.
Another claim with reference:
For any plane graph G, the medial graph of G and the medial graph of the dual graph of G are isomorphic. Conversely, for any 4-regular plane graph H, the only two plane graphs with medial graph H are dual to each other.
One can decide if a planar graph is dual to another by checking if the medial graphs are isomorphic.
The two Wikipedia claims mean graph isomorphism is NP-complete, which is unlikely collapse.
Q1 What is wrong with this?