Advertisement

Branching in Digraphs with Many and Few Leaves: Structural and Algorithmic Results

  • Jørgen Bang-Jensen
  • Gregory GutinEmail author
Chapter
  • 847 Downloads
Part of the Springer Optimization and Its Applications book series (SOIA, volume 139)

Abstract

A subgraph T of a digraph D is called an out-tree if T is an oriented tree with just one vertex s of in-degree zero. A spanning out-tree is called an out-branching. A vertex x of an out-branching B is called a leaf if the out-degree of x is zero. This is a survey on out-branchings with minimum and maximum number of leaves covering both structural and algorithmic results.

Keywords

Algorithmic Results Acyclic Digraph General Digraphs Input Digraph fixed-parameter Tractable (FPT) 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

The research of Jørgen Bang-Jensen was supported by the Danish research council, grant number 1323-00178B. The research of Gregory Gutin was supported in part by Royal Society Wolfson Research Merit Award.

References

  1. 1.
    N. Alon, F. Fomin, G. Gutin, M. Krivelevich, S. Saurabh, Parameterized algorithms for directed maximum leaf problems, in Proceedings of ICALP 2007. Lecture Notes in Computer Science, vol. 4596 (2007), pp. 352–362CrossRefzbMATHGoogle Scholar
  2. 2.
    N. Alon, F. Fomin, G. Gutin, M. Krivelevich, S. Saurabh, Spanning directed trees with many leaves. SIAM J. Discret. Math. 23, 466–476 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    J. Bang-Jensen, Edge-disjoint in- and out-branchings in tournaments and related path problems. J. Combin. Theory Ser. B 51(1), 1–23 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    J. Bang-Jensen, G. Gutin, Digraphs: Theory, Algorithms and Applications, 2nd edn. (Springer, Berlin, 2009)CrossRefzbMATHGoogle Scholar
  5. 5.
    J. Bang-Jensen, G. Gutin, Out-branchings with extremal number of leaves. Ramanujan Math. Soc. Lect. Notes 13, 91–99 (2010)MathSciNetzbMATHGoogle Scholar
  6. 6.
    J. Bang-Jensen, S. Simonsen, Arc-disjoint paths and trees in 2-regular digraphs. Discret. Appl. Math. 161(16–17), 2724–2730 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    J. Bang-Jensen, A. Yeo, The minimum spanning strong subdigraph problem is fixed parameter tractable. Discret. Appl. Math. 156, 2924–2929 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    J. Bang-Jensen, S. Saurabh, S. Simonsen, Parameterized algorithms for non-separating trees and branchings in digraphs. Algorithmica 76(1), 279–296 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    S.J. Berkowitz, On computing the determinant in small parallel time using a small number of processors. Inf. Process. Lett. 18 147–150 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    D. Binkele-Raible, H. Fernau, F.V. Fomin, D. Lokshtanov, S. Saurabh, Y. Villanger, Kernel(s) for problems with n kernel: on out-trees with many leaves. ACM Trans. Algorithms 9(4) (2011), article 39Google Scholar
  11. 11.
    A. Björklund, P. Kaski, L. Kowalik, Constrained multilinear detection and generalized graph motifs. Algorithmica 74(2), 947–967 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    A. Björklund, V. Kamat, L. Kowalik, M. Zehavi, Spotting trees with few leaves. SIAM J. Discret. Math. 31(2), 687–713 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    A. Björklund, P. Kaski, I. Koutis, Directed hamiltonicity and out-branchings via generalized Laplacians, in Automata, Languages and Programming, 44th International Colloquium, ICALP 2017. Leibniz International Proceedings in Informatics (LIPIcs), vol. 80 (2017), pp. 91:1–91:14Google Scholar
  14. 14.
    P. Bonsma, F. Dorn, Tight bounds and a fast FPT algorithm for directed Max-Leaf Spanning Tree. J. ACM Trans. Algorithms 7(4), 1–19 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    N. Cohen, F.V. Fomin, G. Gutin, E.J. Kim, S. Saurabh, A. Yeo, Algorithm for finding k-vertex out-trees and its application to k-internal out-branching problem. J. Comput. Syst. Sci. 76, 650–662 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    M. Cygan, F.V. Fomin, L. Kowalik, D. Lokshtanov, D. Marx, M. Pilipczuk, M. Pilipczuk, S. Saurabh, Parameterized Algorithms (Springer, Berlin, 2015)CrossRefzbMATHGoogle Scholar
  17. 17.
    J. Daligault, Combinatorial techniques for parameterized algorithms and kernels, with applications to multicut, PhD thesis, Universite Montpellier II, Montpellier, Herault, 2011Google Scholar
  18. 18.
    J. Daligault, G. Gutin, E.J. Kim, A. Yeo, FPT algorithms and kernels for the directed k-leaf problem. J. Comput. Syst. Sci. 76, 144–152 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    P. Dankelmann, G. Gutin, E.J. Kim, On complexity of minimum leaf out-branching problem. Discret. Appl. Math. 157, 3000–3004 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    A. Demers, A. Downing, minimum leaf spanning tree. US Patent no. 6,105,018, August 2000Google Scholar
  21. 21.
    R.P. Dilworth, A decomposition theorem for partially ordered sets. Ann. Math. 51, 161–166 (1950)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    J. Edmonds, Edge-disjoint branchings, in Combinatorial Algorithms, ed. by B. Rustin (Academic Press, Cambridge, 1973), pp. 91–96Google Scholar
  23. 23.
    F.V. Fomin, F. Grandoni, D. Lokshtanov, S. Saurabh, Sharp separation and applications to exact and parameterized algorithms. Algorithmica 63, 692–706 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    F.V. Fomin, S. Gaspers, S. Saurabh, S. Thomassé, A linear vertex kernel for maximum internal spanning tree. J. Comput. Syst. Sci. 79, 1–6 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    T. Gallai, A.N. Milgram, Verallgemeinerung eines graphentheoretischen Satzes von Rédei. Acta Sci. Math. Szeged 21, 181–186 (1960)MathSciNetzbMATHGoogle Scholar
  26. 26.
    R. Ganian, P. Hlineny, A. Langer, J. Obdrzalek, P. Rossmanith, Digraph width measures in parameterized algorithmics. Discret. Appl. Math. 168, 88–107 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    M.R. Garey, D.S. Johnson, Computers and Intractability (W.H. Freeman and Co., San Francisco, 1979)zbMATHGoogle Scholar
  28. 28.
    G. Gutin, I. Razgon, E.J. Kim, Minimum leaf out-branching problems, in AAIM’08. Lecture Notes in Computer Science, vol. 5034 (2008), pp. 235–246Google Scholar
  29. 29.
    G. Gutin, I. Razgon, E.J. Kim, Minimum leaf out-branching and other problems. Theor. Comput. Sci. 410, 4571–4579 (2009)CrossRefzbMATHGoogle Scholar
  30. 30.
    G. Gutin, F. Reidl, M. Wahlström, k-distinct in- and out-branchings in digraphs. J. Comput. Syst. Sci. 95, 86–97 (2018)Google Scholar
  31. 31.
    G. Gutin, F. Reidl, M. Wahlström, M. Zehavi, Designing deterministic polynomial-space algorithms by color-coding multivariate polynomials. J. Comput. Syst. Sci. 95, 69–85 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    T. Johnson, N. Robertson, P.D. Seymour, R. Thomas, Directed tree-width. J. Combin. Theory Ser. B 82, 138–154 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    J. Kneis, A. Langer, P. Rossmanith, A new algorithm for finding trees with many leaves, in ISAAC 2008. Lecture Notes in Computer Science, vol. 5369 (2008), pp. 270–281CrossRefzbMATHGoogle Scholar
  34. 34.
    M. Las Vergnas, Sur les arborescences dans un graphe orienté. Discret. Math. 15, 27–39 (1976)CrossRefzbMATHGoogle Scholar
  35. 35.
    W. Li, Y. Cao, J. Chen, J. Wang, Deeper local search for parameterized and approximation algorithms for maximum internal spanning tree. Inf. Comput. 252, 187–200 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    E. Prieto, C. Sloper, Reducing to independent set structure – the case of k-internal spanning tree. Nord. J. Comput. 12, 308–318 (2005)MathSciNetzbMATHGoogle Scholar
  37. 37.
    H. Shachnai, M. Zehavi, Representative families: a unified tradeoff-based approach. J. Comput. Syst. Sci. 82, 488–502 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    S. Thomassé, Covering a strong digraph by α − 1 disjoint paths: a proof of Las Vergnas’ conjecture. J. Combin. Theory Ser. B 83, 331–333 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    M. Zehavi, Mixing color coding-related techniques, in Algorithms - ESA 2015 - 23rd Annual European Symposium. Lecture Notes in Computer Science, vol. 9294 (2015), pp. 1037–1049CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.IMADA, University of Southern DenmarkOdense MDenmark
  2. 2.Department of Computer Science, Royal HollowayUniversity of LondonEgham, SurreyUK

Personalised recommendations