The definition of Book Embedding on Wikipedia: "A book embedding is a generalization of planar embedding of a graph to embeddings into a book, a collection of half-planes all having the same line as their boundary. The vertices of the graph are required to lie on this boundary line, and the edges are required to stay within a single half-plane". Full definition is at (http://en.wikipedia.org/wiki/Book_embedding)
Planar embeddings have geometric duals, that has numerous algorithmic application. One of the very nice property of duality of planar graphs is that if $T$ is a spanning tree of a planar grpah $G$, then the set of edges not in $T$ in the dual is a spanning tree of the dual $G^*$. Is there any kind of similar geometric dual for book embeddings? Is there a similar property between the spanning tree of primal and dual graphs as in the planar case?