Skip to main content
SpringerLink
Log in

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

  • Chapter
  • First Online:
Optimization Problems in Graph Theory

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 139))

  • 1476 Accesses

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 79.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 99.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 129.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    We removed results on lower bounds on the maximum number of leaves in an out-branching as it does not seem to be of interest any longer, and added new structural and algorithmic results, especially on fixed-parameter tractable algorithms.

  2. 2.

    The algorithms of [14] and [28] have been significantly improved and we discuss the improvements in the survey.

  3. 3.

    For an excellent introduction to the area of parameterized algorithms and complexity, see the monograph [16] by Cygan et al.

  4. 4.

    For more information of the area of parameterized kernels, see [16].

References

  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. 352–362

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  4. J. Bang-Jensen, G. Gutin, Digraphs: Theory, Algorithms and Applications, 2nd edn. (Springer, Berlin, 2009)

    Book  MATH  Google Scholar 

  5. J. Bang-Jensen, G. Gutin, Out-branchings with extremal number of leaves. Ramanujan Math. Soc. Lect. Notes 13, 91–99 (2010)

    MathSciNet  MATH  Google Scholar 

  6. J. Bang-Jensen, S. Simonsen, Arc-disjoint paths and trees in 2-regular digraphs. Discret. Appl. Math. 161(16–17), 2724–2730 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. J. Bang-Jensen, A. Yeo, The minimum spanning strong subdigraph problem is fixed parameter tractable. Discret. Appl. Math. 156, 2924–2929 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. J. Bang-Jensen, S. Saurabh, S. Simonsen, Parameterized algorithms for non-separating trees and branchings in digraphs. Algorithmica 76(1), 279–296 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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 39

    Google Scholar 

  11. A. Björklund, P. Kaski, L. Kowalik, Constrained multilinear detection and generalized graph motifs. Algorithmica 74(2), 947–967 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. A. Björklund, V. Kamat, L. Kowalik, M. Zehavi, Spotting trees with few leaves. SIAM J. Discret. Math. 31(2), 687–713 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  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. 91:1–91:14

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  16. M. Cygan, F.V. Fomin, L. Kowalik, D. Lokshtanov, D. Marx, M. Pilipczuk, M. Pilipczuk, S. Saurabh, Parameterized Algorithms (Springer, Berlin, 2015)

    Book  MATH  Google Scholar 

  17. J. Daligault, Combinatorial techniques for parameterized algorithms and kernels, with applications to multicut, PhD thesis, Universite Montpellier II, Montpellier, Herault, 2011

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  19. P. Dankelmann, G. Gutin, E.J. Kim, On complexity of minimum leaf out-branching problem. Discret. Appl. Math. 157, 3000–3004 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. A. Demers, A. Downing, minimum leaf spanning tree. US Patent no. 6,105,018, August 2000

    Google Scholar 

  21. R.P. Dilworth, A decomposition theorem for partially ordered sets. Ann. Math. 51, 161–166 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  22. J. Edmonds, Edge-disjoint branchings, in Combinatorial Algorithms, ed. by B. Rustin (Academic Press, Cambridge, 1973), pp. 91–96

    Google Scholar 

  23. F.V. Fomin, F. Grandoni, D. Lokshtanov, S. Saurabh, Sharp separation and applications to exact and parameterized algorithms. Algorithmica 63, 692–706 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  25. T. Gallai, A.N. Milgram, Verallgemeinerung eines graphentheoretischen Satzes von Rédei. Acta Sci. Math. Szeged 21, 181–186 (1960)

    MathSciNet  MATH  Google Scholar 

  26. R. Ganian, P. Hlineny, A. Langer, J. Obdrzalek, P. Rossmanith, Digraph width measures in parameterized algorithmics. Discret. Appl. Math. 168, 88–107 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  27. M.R. Garey, D.S. Johnson, Computers and Intractability (W.H. Freeman and Co., San Francisco, 1979)

    MATH  Google Scholar 

  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. 235–246

    Google Scholar 

  29. G. Gutin, I. Razgon, E.J. Kim, Minimum leaf out-branching and other problems. Theor. Comput. Sci. 410, 4571–4579 (2009)

    Article  MATH  Google Scholar 

  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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  32. T. Johnson, N. Robertson, P.D. Seymour, R. Thomas, Directed tree-width. J. Combin. Theory Ser. B 82, 138–154 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  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. 270–281

    Article  MATH  Google Scholar 

  34. M. Las Vergnas, Sur les arborescences dans un graphe orienté. Discret. Math. 15, 27–39 (1976)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  36. E. Prieto, C. Sloper, Reducing to independent set structure – the case of k-internal spanning tree. Nord. J. Comput. 12, 308–318 (2005)

    MathSciNet  MATH  Google Scholar 

  37. H. Shachnai, M. Zehavi, Representative families: a unified tradeoff-based approach. J. Comput. Syst. Sci. 82, 488–502 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. 1037–1049

    Chapter  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

  1. IMADA, University of Southern Denmark, Odense M, Denmark

    Jørgen Bang-Jensen

  2. Department of Computer Science, Royal Holloway, University of London, Egham, Surrey, UK

    Gregory Gutin

Authors
  1. Jørgen Bang-Jensen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gregory Gutin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gregory Gutin .

Editor information

Editors and Affiliations

  1. Department of Information Systems and Decision Science, Merrick School of Business, University of Baltimore, Baltimore, MD, USA

    Boris Goldengorin

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Bang-Jensen, J., Gutin, G. (2018). Branching in Digraphs with Many and Few Leaves: Structural and Algorithmic Results. In: Goldengorin, B. (eds) Optimization Problems in Graph Theory. Springer Optimization and Its Applications, vol 139. Springer, Cham. https://doi.org/10.1007/978-3-319-94830-0_5

Download citation

Publish with us

Policies and ethics

Search

Navigation