This Article 
 Bibliographic References 
 Add to: 
CSD Homomorphisms between Phylogenetic Networks
July-Aug. 2012 (vol. 9 no. 4)
pp. 1128-1138
S. J. Willson, Dept. of Math., Iowa State Univ., Ames, IA, USA
Since Darwin, species trees have been used as a simplified description of the relationships which summarize the complicated network N of reality. Recent evidence of hybridization and lateral gene transfer, however, suggest that there are situations where trees are inadequate. Consequently it is important to determine properties that characterize networks closely related to N and possibly more complicated than trees but lacking the full complexity of N. A connected surjective digraph map (CSD) is a map f from one network N to another network M such that every arc is either collapsed to a single vertex or is taken to an arc, such that f is surjective, and such that the inverse image of a vertex is always connected. CSD maps are shown to behave well under composition. It is proved that if there is a CSD map from N to M, then there is a way to lift an undirected version of M into N, often with added resolution. A CSD map from N to M puts strong constraints on N. In general, it may be useful to study classes of networks such that, for any N, there exists a CSD map from N to some standard member of that class.

[1] H.-J. Bandelt and A. Dress, "Split Decomposition: A New and Useful Approach to Phylogenetic Analysis of Distance Data," Molecular Phylogenetics and Evolution, vol. 1, pp. 242-252, 1992.
[2] M. Baroni, C. Semple, and M. Steel, "A Framework for Representing Reticulate Evolution," Annals of Combinatorics, vol. 8, pp. 391-408, 2005.
[3] P. Buneman, "The Recovery of Trees from Measures of Dissimilarity," Mathematics in the Archaeological and Historical Sciences, F.R. Hodson, D.G. Kendall, and P. Tautu, eds., pp. 387-395, Edinburgh Univ. Press, 1971.
[4] G. Cardona, M. Llabrés, F. Rosselló, and G. Valiente, "A Distance Metric for a Class of Tree-Sibling Phylogenetic Networks," Bioinformatics, vol. 24, pp. 1481-1488, 2008.
[5] G. Cardona, F. Rosselló, and G. Valiente, "Comparison of Tree-Child Phylogenetic Networks," IEEE/ACM Trans. Computational Biology and Bioinformatics, vol. 6, no. 4, pp. 552-569, Oct.-Dec. 2009.
[6] T. Dagan, Y. Artzy-Randrup, and W. Martin, "Modular Networks and Cumulative Impact of Lateral Transfer in Prokaryote Genome Evolution," Proc. Nat'l Academy of Sciences USA, vol. 105, pp. 10039-10044, 2008.
[7] A. Daneshgar, H. Hajiabolhassan, and N. Hamedazimi, "On Connected Colourings of Graphs," Ars Combinatoria, vol. 89, pp. 115-126, 2008.
[8] W.F. Doolittle and E. Bapteste, "Pattern Pluralism and the Tree of Life Hypothesis," Proc. Nat'l Academy of Sciences USA, vol. 104, pp. 2043-2049, 2007.
[9] A. Dress, V. Moulton, M. Steel, and T. Wu, "Species, Clusters and the "Tree of Life": A Graph-Theoretic Perspective," J. Theoretical Biology, vol. 265, no. 4, pp. 535-542, 2010.
[10] D. Gusfield, S. Eddhu, and C. Langley, "Optimal, Efficient Reconstruction of Phylogenetic Networks with Constrained Recombination," J. Bioinformatics and Computational Biology, vol. 2, pp. 173-213, 2004.
[11] G. Hahn and C. Tardif, "Graph Homomorphisms: Structure and Symmetry," Graph Symmetry: Algebraic Methods and Applications, vol. 497, pp. 107-166, 1997.
[12] P. Hell and J. Nešetřil, Graphs and Homomorphisms. Oxford Univ. Press, 2004.
[13] L.J.J. van Iersel, J.C.M. Keijsper, S.M. Kelk, L. Stougie, F. Hagen, and T. Boekhout, "Constructing Level-2 Phylogenetic Networks from Triplets," IEEE/ACM Trans. Computational Biology and Bioinformatics, vol. 6, no. 43, pp. 667-681, Oct.-Dec. 2009.
[14] B.M.E. Moret, L. Nakhleh, T. Warnow, C.R. Linder, A. Tholse, A. Padolina, J. Sun, and R. Timme, "Phylogenetic Networks: Modeling, Reconstructibility, and Accuracy," IEEE/ACM Trans. Computational Biology and Bioinformatics, vol. 1, no. 1, pp. 13-23, Jan.-Mar. 2004.
[15] D.A. Morrison, "Phylogenetic Networks in Systematic Biology (and elsewhere)," Research Advances in Systematic Biology, R. Mohan, ed., pp. 1-48, Global Research Network, Trivandrum, India, 2010.
[16] L. Nakhleh, T. Warnow, and C.R. Linder, "Reconstructing Reticulate Evolution in Species-Theory and Practice," Proc. Eighth Ann. Int'l Conf. Computational Molecular Biology (RECOMB '04), pp. 337-346, 2004.
[17] C. Semple and M. Steel, Phylogenetics. Oxford Univ. Press, 2003.
[18] L. Wang, K. Zhang, and L. Zhang, "Perfect Phylogenetic Networks with Recombination," J. Computational Biology, vol. 8, pp. 69-78, 2001.

Index Terms:
molecular biophysics,complex networks,evolution (biological),genetics,graph theory,network classes,CSD homomorphisms,phylogenetic networks,species trees,reality network,hybridization,gene transfer,connected surjective digraph map,Vegetation,Phylogeny,Bioinformatics,Computational biology,Topology,Kernel,Image edge detection,homomorphism.,Digraph,network,connected,hybrid,phylogeny
S. J. Willson, "CSD Homomorphisms between Phylogenetic Networks," IEEE/ACM Transactions on Computational Biology and Bioinformatics, vol. 9, no. 4, pp. 1128-1138, July-Aug. 2012, doi:10.1109/TCBB.2012.52
Usage of this product signifies your acceptance of the Terms of Use.