A Characterization of Horoidal Digraphs

In this paper we investigate the upward embedding problem on the horizontal torus. The digraphs that admit upward embedding on this surface are called horoidal digraphs. We shall characterize the horoidal digraphs, combinatorially. Then, we construct a new digraph from an arbitrary digraph in such a way that the new digraph has an upward embedding on sphere if and only if it is horoidal. By using these constructed digraphs, we show that the decision problem whether a digraph has an upward embedding on the horizontal torus is NP-Complete.

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