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.
2.
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.
3.
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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