# Reconstructing Phylogenetic Level-1 Networks from Nondense Binet and Trinet Sets

Algorithmica, Sep 2015

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.

This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1007%2Fs00453-015-0069-8.pdf

Katharina T. Huber, Leo van Iersel, Vincent Moulton, Celine Scornavacca, Taoyang Wu. Reconstructing Phylogenetic Level-1 Networks from Nondense Binet and Trinet Sets, Algorithmica, 2017, 173-200, DOI: 10.1007/s00453-015-0069-8