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 level-1 phylogenetic network displaying a given set mathbb {T} of binary binets or trinets over a taxon set X, and constructing such a network whenever it exists. We show that this is NP-hard for trinets but polynomial-time solvable for binets. Moreover, we show that the problem is still polynomial-time solvable for inputs consisting of binets and trinets as long as the cycles in the trinets have size three. Finally, we present an O(3^{|X|} poly(|X|)) time algorithm for general sets of binets and trinets. The latter two algorithms generalise to instances containing level-1 networks with arbitrarily many leaves, and thus provide some of the first supernetwork algorithms for computing networks from a set of rooted phylogenetic networks.
Highlights
A key problem in biology is to reconstruct the evolutionary history of a set of taxa using data such as DNA sequences or morphological features
We have presented an exponential time algorithm for determining whether or not an arbitrary set of binets and trinets is displayed by a level-1 network, shown that this problem is NP-hard, and given some polynomial time algorithms for solving it in certain special instances
It would be interesting to know whether other special instances are solvable in polynomial time (for example, when either S1(x, y; z)type or S2(x; y; z)-type trinets are excluded)
Summary
A key problem in biology is to reconstruct the evolutionary history of a set of taxa using data such as DNA sequences or morphological features. It has been observed that the set of triplets displayed by a level-1 network does not necessarily provide all of the information required to uniquely define or encode the network [5] Motivated by this observation, in [11] an algorithm was developed for constructing level-1 networks from a network analogue of triplets: rooted binary networks with three leaves, or trinets. 5 we show that for the special instance where each cycle in the input trinets has size three our exponential-time algorithm is guaranteed to work in polynomial time This is still the case when the input consists of binary level-1 networks with arbitrarily many leaves as long as all their cycles have length three.
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