In an upward embedding, the sets of incoming and outgoing edges incident to each vertex are contiguous in the cyclic ordering of the edges at the vertex.
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In fact, cyclic ordering is NP-complete, since it solves 3SAT . This is in stark contrast with the recognition problem for linear orders, which can be solved in linear time.
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In some applications it may make sense to distinguish subtypes of unrooted binary trees : a planar embedding of the tree may be fixed by specifying a cyclic ordering for the edges at each vertex, making it into a plane tree.
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