I'm computer scientist and in one of my researches I'm facing this question: if I have a planar graph that admits an Euler path (i.e. has 0 or 2 odd degree vertices, as Euler's theorem says), then his dual graph also admits an Euler path? And its opposite: if I have a graph that doesn't admits an Euler path, then its dual also doesn't?
This is important because the algorithms that I use to create dual graphs are very computationally costly, so it takes a long time just to check the Euler paths in a graph and in its dual.
I'll be thankful if you can guide me to an answer or even a textbook/paper so I can read about it! Thanks a lot!