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A digraph embedding (Tutte called it alternating dimap) is an Eulerian embedding D which is oriented so that the edges around each vertex are directed alternately into and out of that vertex.The directed genus distribution of the digraph D is the sequence g0(D), g1(D), g2(D), …,where gi(D) is the number of cellular directed embeddings of D on the surface Si.A well-known conjecture in topological graph theory, which says that all genus distributions of digraphs (graphs) are log-concave.