k-Distinct In- and Out-Branchings in Digraphs

Authors Gregory Gutin, Felix Reidl, Magnus Wahlström

Thumbnail PDF


  • Filesize: 0.57 MB
  • 13 pages

Document Identifiers

Author Details

Gregory Gutin
Felix Reidl
Magnus Wahlström

Cite AsGet BibTex

Gregory Gutin, Felix Reidl, and Magnus Wahlström. k-Distinct In- and Out-Branchings in Digraphs. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 58:1-58:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)


An out-branching and an in-branching of a digraph D are called k-distinct if each of them has k arcs absent in the other. Bang-Jensen, Saurabh and Simonsen (2016) proved that the problem of deciding whether a strongly connected digraph D has k-distinct out-branching and in-branching is fixed-parameter tractable (FPT) when parameterized by k. They asked whether the problem remains FPT when extended to arbitrary digraphs. Bang-Jensen and Yeo (2008) asked whether the same problem is FPT when the out-branching and in-branching have the same root. By linking the two problems with the problem of whether a digraph has an out-branching with at least k leaves (a leaf is a vertex of out-degree zero), we first solve the problem of Bang-Jensen and Yeo (2008). We then develop a new digraph decomposition called the rooted cut decomposition and using it we prove that the problem of Bang-Jensen et al. (2016) is FPT for all digraphs. We believe that the rooted cut decomposition will be useful for obtaining other results on digraphs.
  • Digraphs
  • Branchings
  • Decompositions
  • FPT algorithms


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. N. Alon, F. V. Fomin, G. Gutin, M. Krivelevich, and S. Saurabh. Spanning directed trees with many leaves. SIAM Journal on Discrete Mathematics, 23(1):466-476, 2009. Google Scholar
  2. J. Bang-Jensen. Edge-disjoint in- and out-branchings in tournaments and related path problems. Journal of Combinatorial Theory, Series B, 51(1):1-23, 1991. Google Scholar
  3. J. Bang-Jensen and G. Gutin. Digraphs: Theory, Algorithms and Applications. Springer, 2002. Google Scholar
  4. J. Bang-Jensen and J. Huang. Quasi-transitive digraphs. Journal of Graph Theory, 20:141-161, 1995. Google Scholar
  5. J. Bang-Jensen and J. Huang. Arc-disjoint in- and out-branchings with the same root in locally semicomplete digraphs. Journal of Graph Theory, 77:278-298, 2014. Google Scholar
  6. J. Bang-Jensen, S. Saurabh, and S. Simonsen. Parameterized algorithms for non-separating trees and branchings in digraphs. Algorithmica, 76(1):279-296, 2016. Google Scholar
  7. J. Bang-Jensen and S. Simonsen. Arc-disjoint paths and trees in 2-regular digraphs. Discrete Applied Mathematics, 161:2724-2730, 2013. Google Scholar
  8. J. Bang-Jensen, S. Thomassé, and A. Yeo. Small degree out-branchings. Journal of Graph Theory, 42(4):297-307, 2003. Google Scholar
  9. J. Bang-Jensen and A. Yeo. The minimum spanning strong subdigraph problem is fixed parameter tractable. Discrete Applied Mathematics, 156:2924-2929, 2008. Google Scholar
  10. K. Bérczi, S. Fujishige, and N. Kamiyama. A linear-time algorithm to find a pair of arc-disjoint spanning in-arborescence and out-arborescence in a directed acyclic graph. Information Processing Letters, 109(23-24):1227-1231, 2009. Google Scholar
  11. M. Cygan, F.V. Fomin, L. Kowalik, D. Lokshtanov, D. Marx, M. Pilipczuk, M. Pilipczuk, and S. Saurabh. Parameterized Algorithms. Springer, 2015. Google Scholar
  12. J. Daligault, G. Gutin, E. J. Kim, and A. Yeo. FPT algorithms and kernels for the directed k-leaf problem. Journal of Computer and System Sciences, 76(2):144-152, 2010. Google Scholar
  13. R.G. Downey and M.R. Fellows. Fundamentals of Parameterized Complexity. Springer, 2013. Google Scholar
  14. J. Edmonds. Optimum branchings. Journal of Research of the National Bureau of Standards, Section B, 71B:233-240, 1967. Google Scholar
  15. J. Edmonds. Edge-disjoint branchings. In B. Rustin, editor, Combinatorial Algorithms, pages 91-96. Academic Press, 1973. Google Scholar
  16. A. Fradkin and P.D. Seymour. Tournament pathwidth and topological containment. Journal of Combinatorial Theory, Series B, 103(3):374-384, 2013. Google Scholar
  17. K. Kawarabayashi and S. Kreutzer. The directed grid theorem. In Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, STOC 2015, pages 655-664, 2015. Google Scholar
  18. I. Kim and P. D. Seymour. Tournament minors. Journal of Combinatorial Theory, Series B, 112:138-153, 2015. Google Scholar
  19. J. Kneis, A. Langer, and P. Rossmanith. A new algorithm for finding trees with many leaves. Algorithmica, 61(4):882-897, 2011. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail