





Andreas Gunawan,
Bhaskar DasGupta and
Louxin Zhang. A decomposition theorem and two algorithms for reticulationvisible networks. In Information and Computation, Vol. 252:161175, 2017. Keywords: cluster containment, explicit network, from clusters, from network, from rooted trees, phylogenetic network, phylogeny, polynomial, reticulationvisible network, tree containment.. Note: https://www.cs.uic.edu/~dasgupta/resume/publ/papers/Infor_Comput_IC4848_final.pdf.






Andreas Gunawan,
Bhaskar DasGupta and
Louxin Zhang. Locating a Tree in a ReticulationVisible Network in Cubic Time. In RECOMB2016, Vol. 9649:266 of LNBI, Springer, 2016. Keywords: cluster containment, explicit network, from clusters, from network, from rooted trees, phylogenetic network, phylogeny, polynomial, reticulationvisible network, tree containment. Note: http://arxiv.org/abs/1507.02119.






Mareike Fischer,
Leo van Iersel,
Steven Kelk and
Celine Scornavacca. On Computing The Maximum Parsimony Score Of A Phylogenetic Network. In SIDMA, Vol. 29(1):559585, 2015. Keywords: APX hard, cluster containment, explicit network, FPT, from network, from sequences, integer linear programming, level k phylogenetic network, NP complete, parsimony, phylogenetic network, phylogeny, polynomial, Program MPNet, reconstruction, software. Note: http://arxiv.org/abs/1302.2430.






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."






Daniel H. Huson,
Regula Rupp,
Vincent Berry,
Philippe Gambette and
Christophe Paul. Computing Galled Networks from Real Data. In ISMBECCB09, Vol. 25(12):i85i93 of BIO, 2009. Keywords: abstract network, cluster containment, explicit network, FPT, from clusters, from rooted trees, galled network, NP complete, phylogenetic network, phylogeny, polynomial, Program Dendroscope, reconstruction. Note: http://hallirmm.ccsd.cnrs.fr/lirmm00368545/en/.
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"Motivation: Developing methods for computing phylogenetic networks from biological data is an important problem posed by molecular evolution and much work is currently being undertaken in this area. Although promising approaches exist, there are no tools available that biologists could easily and routinely use to compute rooted phylogenetic networks on real datasets containing tens or hundreds of taxa. Biologists are interested in clades, i.e. groups of monophyletic taxa, and these are usually represented by clusters in a rooted phylogenetic tree. The problem of computing an optimal rooted phylogenetic network from a set of clusters, is hard, in general. Indeed, even the problem of just determining whether a given network contains a given cluster is hard. Hence, some researchers have focused on topologically restricted classes of networks, such as galled trees and levelk networks, that are more tractable, but have the practical drawback that a given set of clusters will usually not possess such a representation. Results: In this article, we argue that galled networks (a generalization of galled trees) provide a good tradeoff between level of generality and tractability. Any set of clusters can be represented by some galled network and the question whether a cluster is contained in such a network is easy to solve. Although the computation of an optimal galled network involves successively solving instances of two different NPcomplete problems, in practice our algorithm solves this problem exactly on large datasets containing hundreds of taxa and many reticulations in seconds, as illustrated by a dataset containing 279 prokaryotes. © 2009 The Author(s)."






Luay Nakhleh and
LiSan Wang. Phylogenetic Networks, Trees, and Clusters. In IWBRA05, Vol. 3515:919926 of LNCS, springer, 2005. Keywords: cluster containment, evaluation, from clusters, from network, from rooted trees, phylogenetic network, phylogeny, polynomial, tree child network, tree containment. Note: http://www.cs.rice.edu/~nakhleh/Papers/NakhlehWang.pdf.



Luay Nakhleh and
LiSan Wang. Phylogenetic Networks: Properties and Relationship to Trees and Clusters. In TCSB2, Vol. 3680:8299 of LNCS, springer, 2005. Keywords: cluster containment, evaluation, from clusters, from network, from rooted trees, phylogenetic network, phylogeny, polynomial, tree child network, tree containment. Note: http://www.cs.rice.edu/~nakhleh/Papers/LNCS_TCSB05.pdf.



