Bulletin of Mathematical Biology

, Volume 75, Issue 10, pp 1879–1890 | Cite as

Cherry Picking: A Characterization of the Temporal Hybridization Number for a Set of Phylogenies

  • Peter J. Humphries
  • Simone LinzEmail author
  • Charles Semple
Original Article


Recently, we have shown that calculating the minimum–temporal-hybridization number for a set \({\mathcal{P}}\) of rooted binary phylogenetic trees is NP-hard and have characterized this minimum number when \({\mathcal{P}}\) consists of exactly two trees. In this paper, we give the first characterization of the problem for \({\mathcal{P}}\) being arbitrarily large. The characterization is in terms of cherries and the existence of a particular type of sequence. Furthermore, in an online appendix to the paper, we show that this new characterization can be used to show that computing the minimum–temporal hybridization number for two trees is fixed-parameter tractable.


Cherry Fixed-parameter tractability Phylogenetic network Phylogenetic tree Temporal network 



We thank the referees for their helpful comments. S.L. was supported by a Marie Curie International Outgoing Fellowship within the 7th European Community Framework Programme. C.S. was supported by the New Zealand Marsden Fund and the Allan Wilson Centre for Molecular Ecology and Evolution.

Supplementary material

11538_2013_9874_MOESM1_ESM.pdf (244 kb)
Appendix A.1: Fixed-parameter tractability of 2-Minimum–Temporal Hybridization (PDF 244 kB)


  1. Albrecht, B., Linz, C., & Scornavacca, C. (2012). A first step toward computing all hybridization networks for two rooted binary phylogenetic trees. J. Comput. Biol., 19, 1227–1242. MathSciNetCrossRefGoogle Scholar
  2. Baroni, M., Grünewald, S., Moulton, V., & Semple, C. (2005). Bounding the number of hybridization events for a consistent evolutionary history. J. Math. Biol., 51, 171–182. zbMATHMathSciNetCrossRefGoogle Scholar
  3. Baroni, M., Semple, C., & Steel, M. (2006). Hybrids in real time. Syst. Biol., 44, 46–56. CrossRefGoogle Scholar
  4. Bordewich, M., & Semple, C. (2007). Computing the hybridization number of two phylogenetic trees is fixed-parameter tractable. IEEE/ACM Trans. Comput. Biol. Bioinform., 4, 458–466. CrossRefGoogle Scholar
  5. Bordewich, M., Linz, S., John, K. St., & Semple, C. (2007). A reduction algorithm for computing the hybridization number of two trees. Evol. Bioinform., 3, 86–98. Google Scholar
  6. Cardona, G., Rossello, F., & Valiente, G. (2009). Comparison of tree-child phylogenetic networks. IEEE/ACM Trans. Comput. Biol. Bioinform., 6, 552–569. CrossRefGoogle Scholar
  7. Chen, Z. Z., & Wang, L. (2012). Algorithms for reticulate networks of multiple phylogenetic trees. IEEE/ACM Trans. Comput. Biol. Bioinform., 9, 372–384. CrossRefGoogle Scholar
  8. Chen, Z. Z., & Wang, L. (2013). An ultrafast tool for minimum reticulate networks. J. Comput. Biol., 20, 38–41. MathSciNetCrossRefGoogle Scholar
  9. Collins, J., Linz, S., & Semple, C. (2011). Quantifying hybridization in realistic time. J. Comput. Biol., 18, 1305–1318. MathSciNetCrossRefGoogle Scholar
  10. Gramm, J., & Niedermeier, R. (2003). A fixed-parameter algorithm for minimum quartet inconsistency. J. Comput. Syst. Sci., 67, 723–741. zbMATHMathSciNetCrossRefGoogle Scholar
  11. Gramm, J., Nickelsen, A., & Tantau, T. (2008). Fixed-parameter algorithms in phylogenetics. Comput. J., 51, 79–101. CrossRefGoogle Scholar
  12. Humphries, P. J., Linz, S., & Semple, C. (2013). On the complexity of computing the temporal hybridization number for two phylogenies. Discrete Appl. Math., 161(7–8), 871–880. zbMATHMathSciNetCrossRefGoogle Scholar
  13. Huson, D. H., & Scornavacca, C. (2011). A survey of combinatorial methods for phylogenetic networks. Genome Biol. Evol., 3, 23–35. CrossRefGoogle Scholar
  14. Kelk, S., van Iersel, L., Lekić, N., Linz, S., Scornavacca, C., & Stougie, L. (2012). Cycle killer…qu’est-ce que c’est? On the comparative approximability of hybridization number and directed feedback vertex set. SIAM J. Discrete Math., 26, 1635–1656. zbMATHMathSciNetCrossRefGoogle Scholar
  15. Linz, S., Semple, C., & Stadler, T. (2010). Analyzing and reconstructing reticulation networks under timing constraints. J. Math. Biol., 61, 715–735. zbMATHMathSciNetCrossRefGoogle Scholar
  16. Piovesan, T., & Kelk, S. (2013). A simple fixed parameter tractable algorithm for computing the hybridization number of two (not necessarily binary) trees. IEEE/ACM Trans. Comput. Biol. Bioinform., 10, 18–25. CrossRefGoogle Scholar
  17. van Iersel, L., & Kelk, S. (2011). When two trees go to war. J. Theor. Biol., 269, 245–255. zbMATHCrossRefGoogle Scholar
  18. Whidden, C., Beiko, R. G., & Zeh, N. (2013). Fixed-parameter and approximation algorithms for maximum agreement forests. SIAM J. Comput. 42(4), 1431–1466. zbMATHMathSciNetCrossRefGoogle Scholar
  19. Wu, Y. (2010). Close lower and upper bounds for the minimum reticulate network of multiple phylogenetic trees. Bioinformatics, 26, i140–i148. CrossRefGoogle Scholar
  20. Wu, Y., & Wang, J. (2010). Fast computation of the exact hybridization number of two phylogenetic trees. In LNCS: Vol. 6053. Proceedings of the International Symposium on Bioinformatics Research and Applications (pp. 203–214). CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2013

Authors and Affiliations

  • Peter J. Humphries
    • 1
  • Simone Linz
    • 2
    • 3
    Email author
  • Charles Semple
    • 3
  1. 1.Department of Mathematics and PhysicsNorth Carolina Central UniversityDurhamUSA
  2. 2.Center for BioinformaticsUniversity of TübingenTübingenGermany
  3. 3.Biomathematics Research Centre, Department of Mathematics and StatisticsUniversity of CanterburyChristchurchNew Zealand

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