TITLE: Is deciding if one planar graph is dual to another really NP-hard (Wikipedia claim)?
QUESTION [15 upvotes]: 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?

REPLY [23 votes]: The dual graph and medial graph depend on the choice of an embedding in the plane. The Wikipedia claim seems to be that testing whether there are choices of embeddings for which two graphs are dual is NP-complete.
It looks as though Wikipedia makes a distinction between a planar graph (can be embedded in the plane) and plane graph (is embedded in the plane).
Edit: Actually, later on in the Wikipedia article, there is a reference, to
Angelini, Patrizio; Bläsius, Thomas; Rutter, Ignaz (2014), "Testing mutual duality of planar graphs", International Journal of Computational Geometry and Applications 24 (4): 325–346, arXiv:1303.1640
and the precise statement is indeed as I understood.