
Jesper Jansson and
Andrzej Lingas. Computing the rooted triplet distance between galled trees by counting triangles. In Journal of Discrete Algorithms, Vol. 25:6678, 2014. Keywords: distance between networks, explicit network, from network, galled network, phylogenetic network, phylogeny, polynomial, triplet distance.
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"We consider a generalization of the rooted triplet distance between two phylogenetic trees to two phylogenetic networks. We show that if each of the two given phylogenetic networks is a socalled galled tree with n leaves then the rooted triplet distance can be computed in o(n2.687) time. Our upper bound is obtained by reducing the problem of computing the rooted triplet distance between two galled trees to that of counting monochromatic and almostmonochromatic triangles in an undirected, edgecolored graph. To count different types of colored triangles in a graph efficiently, we extend an existing technique based on matrix multiplication and obtain several new algorithmic results that may be of independent interest: (i) the number of triangles in a connected, undirected, uncolored graph with m edges can be computed in o(m1.408) time; (ii) if G is a connected, undirected, edgecolored graph with n vertices and C is a subset of the set of edge colors then the number of monochromatic triangles of G with colors in C can be computed in o(n2.687) time; and (iii) if G is a connected, undirected, edgecolored graph with n vertices and R is a binary relation on the colors that is computable in O(1) time then the number of Rchromatic triangles in G can be computed in o(n2.687) time. © 2013 Elsevier B.V. All rights reserved."



Matthieu Willems,
Nadia Tahiri and
Vladimir Makarenkov. A new efficient algorithm for inferring explicit hybridization networks following the NeighborJoining principle. In JBCB, Vol. 12(5), 2014. Keywords: explicit network, from distances, heuristic, phylogenetic network, phylogeny, reconstruction.
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"Several algorithms and software have been developed for inferring phylogenetic trees. However, there exist some biological phenomena such as hybridization, recombination, or horizontal gene transfer which cannot be represented by a tree topology. We need to use phylogenetic networks to adequately represent these important evolutionary mechanisms. In this article, we present a new efficient heuristic algorithm for inferring hybridization networks from evolutionary distance matrices between species. The famous NeighborJoining concept and the leastsquares criterion are used for building networks. At each step of the algorithm, before joining two given nodes, we check if a hybridization event could be related to one of them or to both of them. The proposed algorithm finds the exact tree solution when the considered distance matrix is a tree metric (i.e. it is representable by a unique phylogenetic tree). It also provides very good hybrids recovery rates for large trees (with 32 and 64 leaves in our simulations) for both distance and sequence types of data. The results yielded by the new algorithm for real and simulated datasets are illustrated and discussed in detail. © Imperial College Press."



Celine Scornavacca,
Paprotny Wojciech,
Vincent Berry and
Vincent Ranwez. Representing a set of reconciliations in a compact way. In JBCB, Vol. 11(2):1250025, 2013. Keywords: duplication, explicit network, from network, from rooted trees, from species tree, phylogeny, Program GraphDTL, Program TERA, visualization. Note: http://hallirmm.ccsd.cnrs.fr/lirmm00818801.
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"Comparative genomic studies are often conducted by reconciliation analyses comparing gene and species trees. One of the issues with reconciliation approaches is that an exponential number of optimal scenarios is possible. The resulting complexity is masked by the fact that a majority of reconciliation software pick up a random optimal solution that is returned to the enduser. However, the alternative solutions should not be ignored since they tell different stories that parsimony considers as viable as the output solution. In this paper, we describe a polynomial space and time algorithm to build a minimum reconciliation grapha graph that summarizes the set of all most parsimonious reconciliations. Amongst numerous applications, it is shown how this graph allows counting the number of nonequivalent most parsimonious reconciliations. © 2013 Imperial College Press."



Philippe Gambette,
Vincent Berry and
Christophe Paul. Quartets and Unrooted Phylogenetic Networks. In JBCB, Vol. 10(4):1250004, 2012. Keywords: abstract network, circular split system, explicit network, from quartets, level k phylogenetic network, phylogenetic network, phylogeny, polynomial, reconstruction, split, split network. Note: http://hal.archivesouvertes.fr/hal00678046/en/.
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"Phylogenetic networks were introduced to describe evolution in the presence of exchanges of genetic material between coexisting species or individuals. Split networks in particular were introduced as a special kind of abstract network to visualize conflicts between phylogenetic trees which may correspond to such exchanges. More recently, methods were designed to reconstruct explicit phylogenetic networks (whose vertices can be interpreted as biological events) from triplet data. In this article, we link abstract and explicit networks through their combinatorial properties, by introducing the unrooted analog of levelk networks. In particular, we give an equivalence theorem between circular split systems and unrooted level1 networks. We also show how to adapt to quartets some existing results on triplets, in order to reconstruct unrooted levelk phylogenetic networks. These results give an interesting perspective on the combinatorics of phylogenetic networks and also raise algorithmic and combinatorial questions. © 2012 Imperial College Press."



AnChiang Chu,
Jesper Jansson,
Richard Lemence,
Alban Mancheron and
KunMao Chao. Asymptotic Limits of a New Type of Maximization Recurrence with an Application to Bioinformatics. In TAMC12, Vol. 7287:177188 of LNCS, springer, 2012. Keywords: from triplets, galled network, level k phylogenetic network, phylogenetic network. Note: preliminary version.
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"We study the asymptotic behavior of a new type of maximization recurrence, defined as follows. Let k be a positive integer and p k(x) a polynomial of degree k satisfying p k(0) = 0. Define A 0 = 0 and for n ≥ 1, let A n = max 0≤i<n{A i+n kp k(i/n)}. We prove that lim n→∞A n/n n = sup{pk(x)/1x k : 0≤x<1}. We also consider two closely related maximization recurrences S n and S′ n, defined as S 0 = S′ 0 = 0, and for n ≥ 1, S n = max 0≤i<n{S i + i(ni)(ni1)/2} and S′ n = max 0≤i<n{S′ i + ( 3 ni) + 2i( 2 ni) + (ni)( 2 i)}. We prove that lim n→∞ S′n/3( 3 n) = 2(√31)/3 ≈ 0.488033..., resolving an open problem from Bioinformatics about rooted triplets consistency in phylogenetic networks. © 2012 SpringerVerlag."



Jesper Jansson and
Andrzej Lingas. Computing the rooted triplet distance between galled trees by counting triangles. In CPM12, Vol. 7354:385398 of LNCS, springer, 2012. Keywords: distance between networks, explicit network, from network, galled tree, phylogenetic network, phylogeny, polynomial, triplet distance. Note: http://www.df.lth.se/~jj/Publications/d_rt_for_Galled_Trees5_CPM_2012.pdf.
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"We consider a generalization of the rooted triplet distance between two phylogenetic trees to two phylogenetic networks. We show that if each of the two given phylogenetic networks is a socalled galled tree with n leaves then the rooted triplet distance can be computed in o(n 2.688) time. Our upper bound is obtained by reducing the problem of computing the rooted triplet distance to that of counting monochromatic and almost monochromatic triangles in an undirected, edgecolored graph. To count different types of colored triangles in a graph efficiently, we extend an existing technique based on matrix multiplication and obtain several new related results that may be of independent interest. © 2012 SpringerVerlag."



Tetsuo Asano,
Jesper Jansson,
Kunihiko Sadakane,
Ryuhei Uehara and
Gabriel Valiente. Faster computation of the Robinson–Foulds distance between phylogenetic networks. In Information Sciences, Vol. 197:7790, 2012. Keywords: distance between networks, explicit network, level k phylogenetic network, phylogenetic network, polynomial, spread.
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"The RobinsonFoulds distance, a widely used metric for comparing phylogenetic trees, has recently been generalized to phylogenetic networks. Given two phylogenetic networks N 1, N 2 with n leaf labels and at most m nodes and e edges each, the RobinsonFoulds distance measures the number of clusters of descendant leaves not shared by N 1 and N 2. The fastest known algorithm for computing the RobinsonFoulds distance between N 1 and N 2 runs in O(me) time. In this paper, we improve the time complexity to O(ne/log n) for general phylogenetic networks and O(nm/log n) for general phylogenetic networks with bounded degree (assuming the word RAM model with a word length of ⌈logn⌉ bits), and to optimal O(m) time for leafouterplanar networks as well as optimal O(n) time for level1 phylogenetic networks (that is, galledtrees). We also introduce the natural concept of the minimum spread of a phylogenetic network and show how the running time of our new algorithm depends on this parameter. As an example, we prove that the minimum spread of a levelk network is at most k + 1, which implies that for one level1 and one levelk phylogenetic network, our algorithm runs in O((k + 1)e) time. © 2012 Elsevier Inc. All rights reserved."



Michel Habib and
ThuHien To. Constructing a Minimum Phylogenetic Network from a Dense Triplet Set. In JBCB, Vol. 10(5):1250013, 2012. Keywords: explicit network, from triplets, level k phylogenetic network, phylogenetic network, phylogeny, polynomial, reconstruction. Note: http://arxiv.org/abs/1103.2266.
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"For a given set L of species and a set T of triplets on L, we seek to construct a phylogenetic network which is consistent with T i.e. which represents all triplets of T. The level of a network is defined as the maximum number of hybrid vertices in its biconnected components. When T is dense, there exist polynomial time algorithms to construct level0,1 and 2 networks (Aho et al., 1981; Jansson, Nguyen and Sung, 2006; Jansson and Sung, 2006; Iersel et al., 2009). For higher levels, partial answers were obtained in the paper by Iersel and Kelk (2008), with a polynomial time algorithm for simple networks. In this paper, we detail the first complete answer for the general case, solving a problem proposed in Jansson and Sung (2006) and Iersel et al. (2009). For any k fixed, it is possible to construct a levelk network having the minimum number of hybrid vertices and consistent with T, if there is any, in time O(T k+1 n⌊4k/3⌋+1). © 2012 Imperial College Press."



Ruogu Sheng and
Sergey Bereg. Approximating Metrics with Planar BoundaryLabeled Phylogenetic Networks. In JBCB, Vol. 10(6):1250017, 2012. Keywords: abstract network, from distances, phylogenetic network, phylogeny, reconstruction.
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"Phylogenetic networks are useful for visualizing evolutionary relationships between species with reticulate events such as hybridizations and horizontal gene transfers. In this paper, we consider the problem of constructing undirected phylogenetic networks that (1) are planar graphs and (2) admit embeddings in the plane where the vertices labeling all taxa are on the boundary of the network. We develop a new algorithm for constructing phylogenetic networks satisfying these constraints. First, we show that only approximate networks can be constructed for some distance matrices with at least five taxa. Then we prove that any fivepoint metric can be represented approximately by a planar boundarylabeled network with guaranteed fit value of 94.79. We extend the networks constructed in the proof to design an algorithm for computing planar boundarylabeled networks for any number of taxa. © 2012 Imperial College Press."



Marc Thuillard and
Vincent Moulton. Identifying and reconstructing lateral transfers from distance matrices by combining the Minimum Contradiction Method and NeighborNet. In JBCB, Vol. 9(4):453470, 2011. Keywords: from distances, lateral gene transfer, minimum contradiction, NeighborNet, phylogenetic network, phylogeny, reconstruction. Note: http://dx.doi.org/10.1142/S0219720011005409, slides available at http://www.newton.ac.uk/programmes/PLG/seminars/062015501.html.
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"Identifying lateral gene transfers is an important problem in evolutionary biology. Under a simple model of evolution, the expected values of an evolutionary distance matrix describing a phylogenetic tree fulfill the socalled Kalmanson inequalities. The Minimum Contradiction method for identifying lateral gene transfers exploits the fact that lateral transfers may generate large deviations from the Kalmanson inequalities. Here a new approach is presented to deal with such cases that combines the NeighborNet algorithm for computing phylogenetic networks with the Minimum Contradiction method. A subset of taxa, prescribed using NeighborNet, is obtained by measuring how closely the Kalmanson inequalities are fulfilled by each taxon. A criterion is then used to identify the taxa, possibly involved in a lateral transfer between nonconsecutive taxa. We illustrate the utility of the new approach by applying it to a distance matrix for Archaea, Bacteria, and Eukaryota. © 2011 Imperial College Press."



Tetsuo Asano,
Jesper Jansson,
Kunihiko Sadakane,
Ryuhei Uehara and
Gabriel Valiente. Faster Computation of the RobinsonFoulds Distance between Phylogenetic Networks. In CPM10, Vol. 6129:190201 of LNCS, springer, 2010. Keywords: distance between networks, explicit network, level k phylogenetic network, phylogenetic network, polynomial, spread. Note: http://hdl.handle.net/10119/9859, slides available at http://cs.nyu.edu/parida/CPM2010/MainPage_files/18.pdf.
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"The RobinsonFoulds distance, which is the most widely used metric for comparing phylogenetic trees, has recently been generalized to phylogenetic networks. Given two networks N1,N2 with n leaves, m nodes, and e edges, the RobinsonFoulds distance measures the number of clusters of descendant leaves that are not shared by N1 and N2. The fastest known algorithm for computing the RobinsonFoulds distance between those networks runs in O(m(m + e)) time. In this paper, we improve the time complexity to O(n(m+ e)/ log n) for general networks and O(nm/log n) for general networks with bounded degree, and to optimal O(m + e) time for planar phylogenetic networks and boundedlevel phylogenetic networks.We also introduce the natural concept of the minimum spread of a phylogenetic network and show how the running time of our new algorithm depends on this parameter. As an example, we prove that the minimum spread of a levelk phylogenetic network is at most k + 1, which implies that for two levelk phylogenetic networks, our algorithm runs in O((k + 1)(m + e)) time. © SpringerVerlag Berlin Heidelberg 2010."



Leo van Iersel,
Steven Kelk and
Matthias Mnich. Uniqueness, intractability and exact algorithms: reflections on levelk phylogenetic networks. In JBCB, Vol. 7(4):597623, 2009. Keywords: explicit network, from triplets, galled tree, level k phylogenetic network, NP complete, phylogenetic network, phylogeny, reconstruction, uniqueness. Note: http://arxiv.org/pdf/0712.2932v2.



Sagi Snir and
Tamir Tuller. The NETHMM approach: Phylogenetic Network Inference by Combining Maximum Likelihood and Hidden Markov Models. In JBCB, Vol. 7(4):625644, 2009. Keywords: explicit network, from sequences, HMM, lateral gene transfer, likelihood, phylogenetic network, phylogeny, statistical model. Note: http://research.haifa.ac.il/~ssagi/published%20papers/SnirNETHMMJBCB2009.pdf.
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"Horizontal gene transfer (HGT) is the event of transferring genetic material from one lineage in the evolutionary tree to a different lineage. HGT plays a major role in bacterial genome diversification and is a significant mechanism by which bacteria develop resistance to antibiotics. Although the prevailing assumption is of complete HGT, cases of partial HGT (which are also named chimeric HGT) where only part of a gene is horizontally transferred, have also been reported, albeit less frequently. In this work we suggest a new probabilistic model, the NETHMM, for analyzing and modeling phylogenetic networks. This new model captures the biologically realistic assumption that neighboring sites of DNA or amino acid sequences are not independent, which increases the accuracy of the inference. The model describes the phylogenetic network as a Hidden Markov Model (HMM), where each hidden state is related to one of the network's trees. One of the advantages of the NETHMM is its ability to infer partial HGT as well as complete HGT. We describe the properties of the NETHMM, devise efficient algorithms for solving a set of problems related to it, and implement them in software. We also provide a novel complementary significance test for evaluating the fitness of a model (NETHMM) to a given dataset. Using NETHMM, we are able to answer interesting biological questions, such as inferring the length of partial HGT's and the affected nucleotides in the genomic sequences, as well as inferring the exact location of HGT events along the tree branches. These advantages are demonstrated through the analysis of synthetical inputs and three different biological inputs. © 2009 Imperial College Press."







HoLeung Chan,
Jesper Jansson,
TakWah Lam and
SiuMing Yiu. Reconstructing an Ultrametric Galled Phylogenetic Network from a Distance Matrix. In JBCB, Vol. 4(4):807832, 2006. Keywords: explicit network, from distances, galled tree, phylogenetic network, phylogeny, polynomial, reconstruction. Note: http://www.df.lth.se/~jj/Publications/dist_ugn7_JBCB2006.pdf.
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"Given a distance matrix M that specifies the pairwise evolutionary distances between n species, the phylogenetic tree reconstruction problem asks for an edgeweighted phylogenetic tree that satisfies M, if one exists. We study some extensions of this problem to rooted phylogenetic networks. Our main result is an O(n2 log n)time algorithm for determining whether there is an ultrametric galled network that satisfies M, and if so, constructing one. In fact, if such an ultrametric galled network exists, our algorithm is guaranteed to construct one containing the minimum possible number of nodes with more than one parent (hybrid nodes). We also prove that finding a largest possible submatrix M′ of M such that there exists an ultrametric galled network that satisfies M′ is NPhard. Furthermore, we show that given an incomplete distance matrix (i.e. where some matrix entries are missing), it is also NPhard to determine whether there exists an ultrametric galled network which satisfies it. © 2006 Imperial College Press."



YingJun He,
Trinh N. D. Huynh,
Jesper Jansson and
WingKin Sung. Inferring Phylogenetic Relationships Avoiding Forbidden Rooted Triplets. In JBCB, Vol. 4(1):5974, 2006. Note: http://www.df.lth.se/~jj/Publications/forb_triplets7_JBCB2006.pdf.
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"To construct a phylogenetic tree or phylogenetic network for describing the evolutionary history of a set of species is a wellstudied problem in computational biology. One previously proposed method to infer a phylogenetic tree/network for a large set of species is by merging a collection of known smaller phylogenetic trees on overlapping sets of species so that no (or as little as possible) branching information is lost. However, little work has been done so far on inferring a phylogenetic tree/network from a specified set of trees when in addition, certain evolutionary relationships among the species are known to be highly unlikely. In this paper, we consider the problem of constructing a phylogenetic tree/network which is consistent with all of the rooted triplets in a given set C and none of the rooted triplets in another given set F. Although NPhard in the general case, we provide some efficient exact and approximation algorithms for a number of biologically meaningful variants of the problem. © Imperial College Press."



Jesper Jansson and
WingKin Sung. Inferring a level1 phylogenetic network from a dense set of rooted triplets. In TCS, Vol. 363(1):6068, 2006. 1 comment Keywords: explicit network, from triplets, galled tree, level k phylogenetic network, phylogenetic network, phylogeny, polynomial, reconstruction. Note: http://www.df.lth.se/~jj/Publications/ipnrt8_TCS2006.pdf.
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"We consider the following problem: Given a set T of rooted triplets with leaf set L, determine whether there exists a phylogenetic network consistent with T, and if so, construct one. We show that if no restrictions are placed on the hybrid nodes in the solution, the problem is trivially solved in polynomial time by a simple sorting networkbased construction. For the more interesting (and biologically more motivated) case where the solution is required to be a level1 phylogenetic network, we present an algorithm solving the problem in O ( T 2) time when T is dense, i.e., when T contains at least one rooted triplet for each cardinality three subset of L. We also give an O ( T 5 / 3)time algorithm for finding the set of all phylogenetic networks having a single hybrid node attached to exactly one leaf (and having no other hybrid nodes) that are consistent with a given dense set of rooted triplets. © 2006 Elsevier B.V. All rights reserved."



Jesper Jansson,
Nguyen Bao Nguyen and
WingKin Sung. Algorithms for Combining Rooted Triplets into a Galled Phylogenetic Network. In SICOMP, Vol. 35(5):10981121, 2006. 1 comment Keywords: approximation, explicit network, from triplets, galled tree, phylogenetic network, phylogeny, polynomial, reconstruction. Note: http://www.df.lth.se/~jj/Publications/triplets_to_gn7_SICOMP2006.pdf.
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"This paper considers the problem of determining whether a given set Τ of rooted triplets can be merged without conflicts into a galled phylogenetic network and, if so, constructing such a network. When the input Τ is dense, we solve the problem in O(Τ) time, which is optimal since the size of the input is Θ(Τ). In comparison, the previously fastest algorithm for this problem runs in O(Τ2) time. We also develop an optimal O(Τ)time algorithm for enumerating all simple phylogenetic networks leaflabeled by L that are consistent with Τ, where L is the set of leaf labels in Τ, which is used by our main algorithm. Next, we prove that the problem becomes NPhard if extended to nondense inputs, even for the special case of simple phylogenetic networks. We also show that for every positive integer n, there exists some set Τ of rooted triplets on n leaves such that any galled network can be consistent with at most 0.4883 ·Τ of the rooted triplets in Τ. On the other hand, we provide a polynomialtime approximation algorithm that always outputs a galled network consistent with at least a factor of 5/12 (> 0.4166) of the rooted triplets in Τ. © 2006 Society for Industrial and Applied Mathematics."







Charles Choy,
Jesper Jansson,
Kunihiko Sadakane and
WingKin Sung. Computing the maximum agreement of phylogenetic networks. In TCS, Vol. 335(1):93107, 2005. Keywords: dynamic programming, FPT, level k phylogenetic network, MASN, NP complete, phylogenetic network, phylogeny. Note: http://www.df.lth.se/~jj/Publications/masn8_TCS2005.pdf.
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"We introduce the maximum agreement phylogenetic subnetwork problem (MASN) for finding branching structure shared by a set of phylogenetic networks. We prove that the problem is NPhard even if restricted to three phylogenetic networks and give an O(n2)time algorithm for the special case of two level1 phylogenetic networks, where n is the number of leaves in the input networks and where N is called a levelf phylogenetic network if every biconnected component in the underlying undirected graph induces a subgraph of N containing at most f nodes with indegree 2. We also show how to extend our technique to yield a polynomialtime algorithm for any two levelf phylogenetic networks N1,N2 satisfying f=O(logn); more precisely, its running time is O(V(N1)·V(N2)·2f1+f2), where V(Ni) and fi denote the set of nodes in Ni and the level of Ni, respectively, for i∈{1,2}. © 2005 Elsevier B.V. All rights reserved."





Trinh N. D. Huynh,
Jesper Jansson,
Nguyen Bao Nguyen and
WingKin Sung. Constructing a Smallest Refining Galled Phylogenetic Network. In RECOMB05, Vol. 3500:265280 of LNCS, springer, 2005. Keywords: from rooted trees, galled tree, NP complete, phylogenetic network, phylogeny, polynomial, Program SPNet, reconstruction. Note: http://www.df.lth.se/~jj/Publications/refining_gn3_RECOMB2005.pdf.





Jesper Jansson,
Nguyen Bao Nguyen and
WingKin Sung. Algorithms for Combining Rooted Triplets into a Galled Phylogenetic Network. In SODA05, Pages 349358, 2005. 1 comment Keywords: approximation, explicit network, from triplets, galled tree, phylogenetic network, phylogeny, polynomial, reconstruction. Note: http://portal.acm.org/citation.cfm?id=1070481.



Charles Choy,
Jesper Jansson,
Kunihiko Sadakane and
WingKin Sung. Computing the maximum agreement of phylogenetic networks. In Proceedings of Computing: the Tenth Australasian Theory Symposium (CATS'04), Vol. 91:134147 of Electronic Notes in Theoretical Computer Science, 2004. Keywords: dynamic programming, FPT, level k phylogenetic network, MASN, NP complete, phylogenetic network, phylogeny. Note: http://www.df.lth.se/~jj/Publications/masn6_CATS2004.pdf.
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"We introduce the maximum agreement phylogenetic subnetwork problem (MASN) of finding a branching structure shared by a set of phylogenetic networks. We prove that the problem is NPhard even if restricted to three phylogenetic networks and give an O(n2)time algorithm for the special case of two level1 phylogenetic networks, where n is the number of leaves in the input networks and where N is called a levelf phylogenetic network if every biconnected component in the underlying undirected graph contains at most f nodes having indegree 2 in N. Our algorithm can be extended to yield a polynomialtime algorithm for two levelf phylogenetic networks N 1,N2 for any f which is upperbounded by a constant; more precisely, its running time is O(V(N1)·V(N 2)·4f), where V(Ni) denotes the set of nodes of Ni. © 2004 Published by Elsevier B.V."



Dan Gusfield,
Satish Eddhu and
Charles Langley. Optimal, Efficient Reconstruction of Phylogenetic Networks with Constrained Recombination. In JBCB, Vol. 2(1):173213, 2004. Keywords: explicit network, from sequences, galled tree, phylogenetic network, phylogeny, recombination, reconstruction. Note: http://wwwcsif.cs.ucdavis.edu/~gusfield/exfinalrec.pdf.
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"A phylogenetic network is a generalization of a phylogenetic tree, allowing structural properties that are not treelike. In a seminal paper, Wang et al.1 studied the problem of constructing a phylogenetic network, allowing recombination between sequences, with the constraint that the resulting cycles must be disjoint. We call such a phylogenetic network a "galledtree". They gave a polynomialtime algorithm that was intended to determine whether or not a set of sequences could be generated on galledtree. Unfortunately, the algorithm by Wang et al.1 is incomplete and does not constitute a necessary test for the existence of a galledtree for the data. In this paper, we completely solve the problem. Moreover, we prove that if there is a galledtree, then the one produced by our algorithm minimizes the number of recombinations over all phylogenetic networks for the data, even allowing multiplecrossover recombinations. We also prove that when there is a galledtree for the data, the galledtree minimizing the number of recombinations is "essentially unique" . We. also note two additional results: first, any set of sequences that can be derived on a galled tree can be derived on a true tree (without recombination cycles), where at most one back mutation per site is allowed; second, the site compatibility problem (which is NPhard in general) can be solved in polynomial time for any set of sequences that can be derived on a galled tree. Perhaps more important than the specific results about galledtrees, we introduce an approach that can be used to study recombination in general phylogenetic networks. This paper greatly extends the conference version that appears in an earlier work.8 PowerPoint slides of the conference talk can be found at our website. © Imperial College Press."



Jesper Jansson and
WingKin Sung. Inferring a level1 phylogenetic network from a dense set of rooted triplets. In COCOON04, Vol. 3106:462471 of LNCS, springer, 2004. 1 comment Keywords: explicit network, from triplets, galled tree, level k phylogenetic network, phylogenetic network, phylogeny, polynomial, reconstruction. Note: http://www.df.lth.se/~jj/Publications/ipnrt6_COCOON2004.pdf.


