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Feb 10, 2016 at 6:20 vote accept joro
Feb 10, 2016 at 4:38 comment added Tyson Williams I wrote the second quote given by the OP. This answer by Jeremy is great. The essential confusion is that people often speak of the dual graph of planar graph. In reality, an embedding of the planar graph is also needed. While it is possible that a particular embedding is clear from context, I get the impression that most people are unaware of this technicality. For more information, carefully read the Wikipedia section Dual_graph#Uniqueness. The image in that section provides a good example.
Feb 9, 2016 at 18:42 comment added Chris Godsil The wikipedia distinction between plane and planar graphs is quite standard.
Feb 9, 2016 at 10:59 history edited Jeremy Rickard CC BY-SA 3.0
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Feb 9, 2016 at 10:08 history edited Jeremy Rickard CC BY-SA 3.0
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Feb 9, 2016 at 10:01 history answered Jeremy Rickard CC BY-SA 3.0