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Rankings of Directed graphs

Please always quote using this URN: urn:nbn:de:0297-zib-3030
  • 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.

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Metadaten
Author:Jan Kratochvil, Zsolt Tuza
Document Type:ZIB-Report
Date of first Publication:1997/07/10
Series (Serial Number):ZIB-Report (SC-97-34)
ZIB-Reportnumber:SC-97-34
Published in:A prel. vers. appeared in: Graph-Theoretic Concepts in Computer Science. Lecture Notes in Computer Science, 1517. Springer, 1998, pp. 114-123. An extended vers. will appear in SIAM Journal on Discrete Mathematics
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