# Why are graph imbeddings defined the way they are?

In my recent question I asked about a proof for the fact that the dual of a dual graph imbedding is equal to the original graph. Thinking about this a little more leads me to wonder why graph imbeddings are defined using rotation systems at all.

It seems that the natural thing to do, if one is asked to precisely define a graph imbedding, is the more geometric approach of imbedding edges as simple curves on a pre-existing surface. Indeed, this is just an abstraction of what one is doing when they draw a graph on a sheet of paper. The other attractive feature of defining imbeddings this way is that it is possible to construct imbeddings where the faces are not all 2-cells, by contrast with the rotation system definition. The downside of this approach is that there are some subtleties in defining the dual graph, as this picture shows: the imbedding on the right fails to have $G^{**} = G$.

On the other hand, it really is easy to show rigorously that the dual graph construction using rotation systems has the property $G^{**} = G$ for an imbedded graph $G$. Is this why combinatorialists usually define graph imbeddings this way? Or are there deeper reasons to avoid the geometric approach?

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