Bulletin of Mathematical Biology

, Volume 73, Issue 10, pp 2322–2338 | Cite as

Restricted Trees: Simplifying Networks with Bottlenecks

  • Stephen J. WillsonEmail author
Open Access
Original Article


Suppose N is a phylogenetic network indicating a complicated relationship among individuals and taxa. Often of interest is a much simpler network, for example, a species tree T, that summarizes the most fundamental relationships. The meaning of a species tree is made more complicated by the recent discovery of the importance of hybridizations and lateral gene transfers. Hence, it is desirable to describe uniform well-defined procedures that yield a tree given a network N.

A useful tool toward this end is a connected surjective digraph (CSD) map φ:NN′ where N′ is generally a much simpler network than N. A set W of vertices in N is “restricted” if there is at most one vertex uW from which there is an arc into W, thus yielding a bottleneck in N. A CSD map φ:NN′ is “restricted” if the inverse image of each vertex in N′ is restricted in N. This paper describes a uniform procedure that, given a network N, yields a well-defined tree called the “restricted tree” of N. There is a restricted CSD map from N to the restricted tree. Many relationships in the tree can be proved to appear also in N.


Digraph Network Tree Connected Hybrid Phylogeny Homomorphism Restricted Phylogenetic network 


  1. Bandelt, H.-J., & Dress, A. (1992). Split decomposition: a new and useful approach to phylogenetic analysis of distance data. Mol. Phylogenet. Evol., 1, 242–252. CrossRefGoogle Scholar
  2. Baroni, M., Semple, C., & Steel, M. (2004). A framework for representing reticulate evolution. Ann. Comb., 8, 391–408. MathSciNetzbMATHCrossRefGoogle Scholar
  3. Baroni, M., Semple, C., & Steel, M. (2006). Hybrids in real time. Syst. Biol., 55, 46–56. CrossRefGoogle Scholar
  4. Cardona, G., Rosselló, F., & Valiente, G. (2009). Comparison of tree-child phylogenetic networks. IEEE/ACM Trans. Comput. Biol. Bioinform., 6(4), 552–569. CrossRefGoogle Scholar
  5. Dagan, T., Artzy-Randrup, Y., & Martin, W. (2008). Modular networks and cumulative impact of lateral transfer in prokaryote genome evolution. Proc. Natl. Acad. Sci. USA, 105, 10039–10044. CrossRefGoogle Scholar
  6. Degnan, J. H., & Rosenberg, N. A. (2006). Discordance of species trees with their most likely gene trees, PLos. Genetics, 2(5), e68. Google Scholar
  7. Doolittle, W. F., & Bapteste, E. (2007). Pattern pluralism and the Tree of Life hypothesis. Proc. Natl. Acad. Sci. USA, 104, 2043–2049. CrossRefGoogle Scholar
  8. Dress, A., Moulton, V., Steel, M., & Wu, T. (2010). Species, clusters and the ‘tree of life’: a graph-theoretic perspective. J. Theoret. Biol., 265(4), 535–542. CrossRefGoogle Scholar
  9. Gusfield, D., Eddhu, S., & Langley, C. (2004). Optimal, efficient reconstruction of phylogenetic networks with constrained recombination. J. Bioinform. Comput. Biol., 2, 173–213. CrossRefGoogle Scholar
  10. Hahn, G., & Tardif, C. (1997). Graph homomorphisms: structure and symmetry. In G. Hahn & G. Sabidussi (Eds.), NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci.: Vol. 497. Graph symmetry: algebraic methods and applications (pp. 107–166). Dordrecht: Kluwer Academic. Google Scholar
  11. Hell, P., & Nešetřil, J. (2004). Graphs and homomorphisms. London: Oxford University Press. zbMATHCrossRefGoogle Scholar
  12. van Iersel, L. J. J., Keijsper, J. C. M., Kelk, S. M., Stougie, L., Hagen, F., & Boekhout, T. (2009). Constructing level-2 phylogenetic networks from triplets. IEEE/ACM Trans. Comput. Biol. Bioinform., 6(43), 667–681. CrossRefGoogle Scholar
  13. Jin, G., Nakhleh, L., Snir, S., & Tuller, T. (2007). Inferring phylogenetic networks by the maximum parsimony criterion: a case study. Mol. Biol. Evol., 24(1), 324–337. CrossRefGoogle Scholar
  14. Moret, B. M. E., Nakhleh, L., Warnow, T., Linder, C. R., Tholse, A., Padolina, A., Sun, J., & Timme, R. (2004). Phylogenetic networks: modeling, reconstructibility, and accuracy. IEEE/ACM Trans. Comput. Biol. Bioinform., 1, 13–23. CrossRefGoogle Scholar
  15. Morrison, D. A. (2009). Phylogenetic networks in systematic biology (and elsewhere). In R. M. Mohan (Ed.), Research advances in systematic biology (Global Research Network, Trivandrum, India) (pp. 1–48). Google Scholar
  16. Nakhleh, L., Warnow, T., & Linder, C. R. (2004). Reconstructing reticulate evolution in species—theory and practice. In P. E. Bourne & D. Gusfield (Eds.), Proceedings of the eighth annual international conference on computational molecular biology (RECOMB’04, 27–31 March 2004, San Diego, California) (pp. 37–346). New York: ACM. Google Scholar
  17. Rosenberg, N., & Tao, R. (2008). Discordance of species trees with their most likely gene trees: the case of five taxa. Syst. Biol., 57(1), 131–140. CrossRefGoogle Scholar
  18. Wang, L., Zhang, K., & Zhang, L. (2001). Perfect phylogenetic networks with recombination. J. Comput. Biol., 8, 69–78. CrossRefGoogle Scholar
  19. Willson, S. J. (2010), Relationships among phylogenetic networks (submitted). arXiv:1005.2108v1 [q-bio.PE].

Copyright information

© The Author(s) 2011

Authors and Affiliations

  1. 1.Department of MathematicsIowa State UniversityAmesUSA

Personalised recommendations