Spanning trees, Euler tours, medial graphs, left-right paths and cycle spaces

Let G = (V, E) be a graph embedded in a surface 2. (Unless otherwise specifically stated, all graphs and surfaces are assumed to be connected and all embeddings are cellular, i.e. every face is homeomorphic to an open disc.) The geometric dual of G is the graph Go = (D, E), where D is the set of faces of G and an edge e E E joins the two faces (not necessarily different) that lie on either side of e in the embedding of G. The usual embedding of Go in ,Y has G as its dual.