Partitioning the arcs of a digraph into a star forest of the underlying graph with prescribed orientation properties

  • Authors:

  • Jørgen Bang-Jensen;Daniel Gonçalves;Anders Yeo

  • Affiliations:

  • Department of Mathematics and Computer Science, University of Southern Denmark, Odense DK-5230, Denmark;LIRMM, CNRS and Université Montpellier 2, France;Department of Computer Science, Royal Holloway, University of London, Egham, Surrey TW20 0EX, United Kingdom

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2013

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Abstract

A star in an undirected graph is a tree in which at most one vertex has degree larger than one. A star forest is a collection of vertex disjoint stars. An out-star (in-star) in a digraph D is a star in the underlying undirected graph of D such that all edges are directed out of (into) the center. The problem of partitioning the edges of the underlying graph of a digraph D into two star forests F"0 and F"1 is known to be NP-complete. On the other hand, with the additional requirement for F"0 and F"1 to be forests of out-stars the problem becomes polynomial (via an easy reduction to 2-SAT). In this article we settle the complexity of problems lying in between these two problems. Namely, we study the complexity of the related problems where we require each F"i to be a forest of stars in the underlying sense and require (in different problems) that in D, F"i is either a forest of out-stars, in-stars, out- or in-stars or just stars in the underlying sense.