Inferring a level-1 phylogenetic network from a dense set of rooted triplets

Abstract. Given a set T of rooted triplets with leaf set L, we consider the problem of determining whether there exists a phylogenetic network consistent with T, and if so, constructing one. If no restrictions are placed on the hybrid nodes in the solution, the problem is trivially solved in polynomial time by a simple sorting network-based construction. For the more interesting (and biologically more motivated) case where the solution is required to be a level-1 phylogenetic network, we present an algorithm solving the problem in O(n6) time when T is dense (i.e., contains at least one rooted triplet for each cardinality three subset of L), where n = |L|. Note that the size of the input is Θ(n3) if T is dense. We also give an O(n5)-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.