









Philippe Gambette,
Leo van Iersel,
Steven Kelk,
Fabio Pardi and
Celine Scornavacca. Do branch lengths help to locate a tree in a phylogenetic network? In BMB, Vol. 78(9):17731795, 2016. Keywords: branch length, explicit network, FPT, from network, from rooted trees, NP complete, phylogenetic network, phylogeny, pseudopolynomial, time consistent network, tree containment, tree sibling network. Note: http://arxiv.org/abs/1607.06285.






Peter J. Humphries,
Simone Linz and
Charles Semple. On the complexity of computing the temporal hybridization number for two phylogenies. In DAM, Vol. 161:871880, 2013. Keywords: agreement forest, APX hard, characterization, from rooted trees, hybridization, NP complete, phylogenetic network, phylogeny, reconstruction, time consistent network. Note: http://ab.inf.unituebingen.de/people/linz/publications/TAFapx.pdf.
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"Phylogenetic networks are now frequently used to explain the evolutionary history of a set of species for which a collection of gene trees, reconstructed from genetic material of different parts of the species' genomes, reveal inconsistencies. However, in the context of hybridization, the reconstructed networks are often not temporal. If a hybridization network is temporal, then it satisfies the time constraint of instantaneously occurring hybridization events; i.e. all species that are involved in such an event coexist in time. Furthermore, although a collection of phylogenetic trees can often be merged into a hybridization network that is temporal, many algorithms do not necessarily find such a network since their primary optimization objective is to minimize the number of hybridization events. In this paper, we present a characterization for when two rooted binary phylogenetic trees admit a temporal hybridization network. Furthermore, we show that the underlying optimization problem is APXhard and, therefore, NPhard. Thus, unless P=NP, it is unlikely that there are efficient algorithms for either computing an exact solution or approximating it within a ratio arbitrarily close to one. © 2012 Elsevier B.V. All rights reserved."



Peter J. Humphries,
Simone Linz and
Charles Semple. Cherry picking: a characterization of the temporal hybridization number for a set of phylogenies. In BMB, Vol. 75(10):18791890, 2013. Keywords: characterization, from rooted trees, hybridization, NP complete, phylogenetic network, phylogeny, reconstruction, time consistent network. Note: http://ab.inf.unituebingen.de/people/linz/publications/CPSpaper.pdf.
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"Recently, we have shown that calculating the minimumtemporalhybridization number for a set P of rooted binary phylogenetic trees is NPhard and have characterized this minimum number when P consists of exactly two trees. In this paper, we give the first characterization of the problem for 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 minimumtemporal hybridization number for two trees is fixedparameter tractable. © 2013 Society for Mathematical Biology."






Gabriel Cardona,
Mercè Llabrés,
Francesc Rosselló and
Gabriel Valiente. Path lengths in treechild time consistent hybridization networks. In Information Sciences, Vol. 180(3):366383, 2010. Keywords: distance between networks, phylogenetic network, phylogeny, time consistent network, tree child network. Note: http://arxiv.org/abs/0807.0087?context=cs.CE.
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"Hybridization networks are representations of evolutionary histories that allow for the inclusion of reticulate events like recombinations, hybridizations, or lateral gene transfers. The recent growth in the number of hybridization network reconstruction algorithms has led to an increasing interest in the definition of metrics for their comparison that can be used to assess the accuracy or robustness of these methods. In this paper we establish some basic results that make it possible the generalization to treechild time consistent (TCTC) hybridization networks of some of the oldest known metrics for phylogenetic trees: those based on the comparison of the vectors of path lengths between leaves. More specifically, we associate to each hybridization network a suitably defined vector of 'splitted' path lengths between its leaves, and we prove that if two TCTC hybridization networks have the same such vectors, then they must be isomorphic. Thus, comparing these vectors by means of a metric for realvalued vectors defines a metric for TCTC hybridization networks. We also consider the case of fully resolved hybridization networks, where we prove that simpler, 'nonsplitted' vectors can be used. © 2009 Elsevier Inc. All rights reserved."



Simone Linz,
Charles Semple and
Tanja Stadler. Analyzing and reconstructing reticulation networks under timing constraints. In JOMB, Vol. 61(5):715737, 2010. Keywords: explicit network, from rooted trees, hybridization, lateral gene transfer, NP complete, phylogenetic network, phylogeny, reconstruction, time consistent network. Note: http://dx.doi.org/10.1007/s002850090319y..
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"Reticulation networks are now frequently used to model the history of life for various groups of species whose evolutionary past is likely to include reticulation events such as horizontal gene transfer or hybridization. However, the reconstructed networks are rarely guaranteed to be temporal. If a reticulation network is temporal, then it satisfies the two biologically motivated timing constraints of instantaneously occurring reticulation events and successively occurring speciation events. On the other hand, if a reticulation network is not temporal, it is always possible to make it temporal by adding a number of additional unsampled or extinct taxa. In the first half of the paper, we show that deciding whether a given number of additional taxa is sufficient to transform a nontemporal reticulation network into a temporal one is an NPcomplete problem. As one is often given a set of gene trees instead of a network in the context of hybridization, this motivates the second half of the paper which provides an algorithm, called TemporalHybrid, for reconstructing a temporal hybridization network that simultaneously explains the ancestral history of two trees or indicates that no such network exists. We further derive two methods to decide whether or not a temporal hybridization network exists for two given trees and illustrate one of the methods on a grass data set. © 2009 The Author(s)."



Miguel Arenas,
Mateus Patricio,
David Posada and
Gabriel Valiente. Characterization of Phylogenetic Networks with NetTest. In BMCB, Vol. 11:268, 2010. Keywords: explicit network, galled tree, phylogenetic network, Program NetTest, software, time consistent network, tree child network, tree sibling network, visualization. Note: http://dx.doi.org/10.1186/1471210511268, software available at http://darwin.uvigo.es/software/nettest/.
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"Background: Typical evolutionary events like recombination, hybridization or gene transfer make necessary the use of phylogenetic networks to properly depict the evolution of DNA and protein sequences. Although several theoretical classes have been proposed to characterize these networks, they make stringent assumptions that will likely not be met by the evolutionary process. We have recently shown that the complexity of simulated networks is a function of the population recombination rate, and that at moderate and large recombination rates the resulting networks cannot be categorized. However, we do not know whether these results extend to networks estimated from real data.Results: We introduce a web server for the categorization of explicit phylogenetic networks, including the most relevant theoretical classes developed so far. Using this tool, we analyzed statistical parsimony phylogenetic networks estimated from ~5,000 DNA alignments, obtained from the NCBI PopSet and Polymorphix databases. The level of characterization was correlated to nucleotide diversity, and a high proportion of the networks derived from these data sets could be formally characterized.Conclusions: We have developed a public web server, NetTest (freely available from the software section at http://darwin.uvigo.es), to formally characterize the complexity of phylogenetic networks. Using NetTest we found that most statistical parsimony networks estimated with the program TCS could be assigned to a known network class. The level of network characterization was correlated to nucleotide diversity and dependent upon the intra/interspecific levels, although no significant differences were detected among genes. More research on the properties of phylogenetic networks is clearly needed. © 2010 Arenas et al; licensee BioMed Central Ltd."



Leo van Iersel,
Charles Semple and
Mike Steel. Locating a tree in a phylogenetic network. In IPL, Vol. 110(23), 2010. Keywords: cluster containment, explicit network, from network, level k phylogenetic network, normal network, NP complete, phylogenetic network, polynomial, regular network, time consistent network, tree child network, tree containment, tree sibling network. Note: http://arxiv.org/abs/1006.3122.
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"Phylogenetic trees and networks are leaflabelled graphs that are used to describe evolutionary histories of species. The Tree Containment problem asks whether a given phylogenetic tree is embedded in a given phylogenetic network. Given a phylogenetic network and a cluster of species, the Cluster Containment problem asks whether the given cluster is a cluster of some phylogenetic tree embedded in the network. Both problems are known to be NPcomplete in general. In this article, we consider the restriction of these problems to several wellstudied classes of phylogenetic networks. We show that Tree Containment is polynomialtime solvable for normal networks, for binary treechild networks, and for levelk networks. On the other hand, we show that, even for treesibling, timeconsistent, regular networks, both Tree Containment and Cluster Containment remain NPcomplete. © 2010 Elsevier B.V. All rights reserved."






Gabriel Cardona,
Mercè Llabrés,
Francesc Rosselló and
Gabriel Valiente. Metrics for phylogenetic networks I: Generalizations of the RobinsonFoulds metric. In TCBB, Vol. 6(1):4661, 2009. Keywords: distance between networks, explicit network, phylogenetic network, phylogeny, time consistent network, tree child network, tripartition distance. Note: http://dx.doi.org/10.1109/TCBB.2008.70.
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"The assessment of phylogenetic network reconstruction methods requires the ability to compare phylogenetic networks. This is the first in a series of papers devoted to the analysis and comparison of metrics for treechild time consistent phylogenetic networks on the same set of taxa. In this paper, we study three metrics that have already been introduced in the literature: the RobinsonFoulds distance, the tripartitions distance and the $mu$distance. They generalize to networks the classical RobinsonFoulds or partition distance for phylogenetic trees. We analyze the behavior of these metrics by studying their least and largest values and when they achieve them. As a byproduct of this study, we obtain tight bounds on the size of a treechild time consistent phylogenetic network. © 2006 IEEE."



Gabriel Valiente. Combinatorial Pattern Matching Algorithms in Computational Biology Using Perl and R. Pages 184208, Taylor & Francis/CRC Press, 2009. Keywords: counting, distance between networks, galled tree, generation, phylogenetic network, phylogeny, survey, time consistent network, tree child network, tree sibling network. Note: http://books.google.fr/books?id=F4YIIUWb7yMC.






Gabriel Cardona,
Mercè Llabrés,
Francesc Rosselló and
Gabriel Valiente. Phylogenetic Networks: Justification, Models, Distances and Algorithms. In VI Jornadas de Matemática Discreta y Algorítmica (JMDA'08), 2008. Keywords: distance between networks, mu distance, phylogenetic network, phylogeny, polynomial, survey, time consistent network, tree child network, tripartition distance, triplet distance. Note: http://bioinfo.uib.es/media/uploaded/jmda2008_submission_611.pdf.






Mihaela Baroni,
Charles Semple and
Mike Steel. Hybrids in Real Time. In Systematic Biology, Vol. 55(1):4656, 2006. Keywords: agreement forest, from rooted trees, phylogenetic network, phylogeny, polynomial, reconstruction, time consistent network. Note: http://www.math.canterbury.ac.nz/~m.steel/Non_UC/files/research/hybrids.pdf.
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"We describe some new and recent results that allow for the analysis and representation of reticulate evolution by nontree networks. In particular, we (1) present a simple result to show that, despite the presence of reticulation, there is always a welldefined underlying tree that corresponds to those parts of life that do not have a history of reticulation; (2) describe and apply new theory for determining the smallest number of hybridization events required to explain conflicting gene trees; and (3) present a new algorithm to determine whether an arbitrary rooted network can be realized by contemporaneous reticulation events. We illustrate these results with examples. Copyright © Society of Systematic Biologists."






Bernard M. E. Moret,
Luay Nakhleh,
Tandy Warnow,
C. Randal Linder,
Anna Tholse,
Anneke Padolina,
Jerry Sun and
Ruth Timme. Phylogenetic Networks: Modeling, Reconstructibility, and Accuracy. In TCBB, Vol. 1(1):1323, 2004. Keywords: distance between networks, evaluation, phylogenetic network, phylogeny, time consistent network, tripartition distance. Note: http://www.cs.rice.edu/~nakhleh/Papers/tcbb04.pdf.





