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RECONSTRUCTING PHYLOGENETIC LEVEL1 NETWORKS FROM NONDENSE BINET AND TRINET SETS
"... Abstract. Binets and trinets are phylogenetic networks with two and three leaves, respectively. Here we consider the problem of deciding if there exists a binary level1 phylogenetic network displaying a given set T of binary binets or trinets over a taxa set X, and constructing such a network whene ..."
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Abstract. Binets and trinets are phylogenetic networks with two and three leaves, respectively. Here we consider the problem of deciding if there exists a binary level1 phylogenetic network displaying a given set T of binary binets or trinets over a taxa set X, and constructing such a network whenever it exists. We show that this is NPhard for trinets but polynomialtime solvable for binets. Moreover, we show that the problem is still polynomialtime solvable for inputs consisting of binets and trinets as long as the cycles in the trinets have size three. Finally, we present an O(3Xpoly(X)) time algorithm for general sets of binets and trinets. The latter two algorithms generalise to instances containing level1 networks with arbitrarily many leaves, and thus provide some of the first supernetwork algorithms for computing networks from a set of rooted phylogenetic networks. 1.
TRINETS ENCODE TREECHILD AND LEVEL2 PHYLOGENETIC NETWORKS
"... Abstract. Phylogenetic networks generalize evolutionary trees, and are commonly used to represent evolutionary histories of species that undergo reticulate evolutionary processes such as hybridization, recombination and lateral gene transfer. Recently, there has been great interest in trying to dev ..."
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Abstract. Phylogenetic networks generalize evolutionary trees, and are commonly used to represent evolutionary histories of species that undergo reticulate evolutionary processes such as hybridization, recombination and lateral gene transfer. Recently, there has been great interest in trying to develop methods to construct rooted phylogenetic networks from triplets, that is rooted trees on three species. However, although triplets determine or encode rooted phylogenetic trees, they do not in general encode rooted phylogenetic networks, which is a potential issue for any such method. Motivated by this fact, Huber and Moulton recently introduced trinets as a natural extension of rooted triplets to networks. In particular, they showed that level1 phylogenetic networks are encoded by their trinets, and also conjectured that all “recoverable ” rooted phylogenetic networks are encoded by their trinets. Here we prove that recoverable binary level2 networks and binary treechild networks are also encoded by their trinets. To do this we prove two decomposition theorems based on trinets which hold for all recoverable binary rooted phylogenetic networks. Our results provide some additional evidence in support of the conjecture that trinets encode all recoverable rooted phylogenetic networks, and could also lead to new approaches to construct phylogenetic networks from trinets. 1.