TY - GEN A1 - Kratochvil, Jan A1 - Tuza, Zsolt T1 - Rankings of Directed graphs N2 - A ranking of a graph is a coloring of the vertex set with positive integers such that on every path connecting two vertices of the same color, there is a vertex of larger color. We consider the directed variant of this problem, where the above condition is imposed only on those paths in which all edges are oriented in the same direction. We show that the ranking number of a directed tree is bounded by that of its longest directed path plus one, and that it can be computed in polynomial time. Unlike the undirected case, however, deciding if the ranking number of a directed (and even of an acyclic directed) graph is bounded by a constant is NP-complete. In fact, the 3-ranking of planar bipartite acyclic digraphs is already hard. T3 - ZIB-Report - SC-97-34 Y1 - 1997 UR - https://opus4.kobv.de/opus4-zib/frontdoor/index/index/docId/303 UR - https://nbn-resolving.org/urn:nbn:de:0297-zib-3030 ER -