Reconstructing Phylogenetic Level-1 Networks from Nondense Binet and Trinet Sets - Huber, Katharina T.; van Iersel, Leo; Moulton, Vincent; Scornavacca, Celine; Wu, Taoyang (Algorithmica, 2017)
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 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 | p o l y ( | 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.