Constructing the simplest possible phylogenetic network from triplets

A phylogenetic network is a directed acyclic graph that visualizes an evolutionary history containing so-called reticulations such as recombinations, hybridizations or lateral gene transfers. Here we consider the construction of a simplest possible phylogenetic network consistent with an input set t, where t contains at least one phylogenetic tree on three leaves (a triplet) for each combination of three taxa. To quantify the complexity of a network we consider both the total number of reticulations and the number of reticulations per biconnected component, called the level of the network. We give polynomial-time algorithms for constructing a level-1 respectively a level-2 network that contains a minimum number of reticulations and is consistent with t (if such a network exists). In addition, we show that if t is precisely equal to the set of triplets consistent with some network, then we can construct such a network with smallest possible level in time o(|t|k+1), if k is a fixed upper bound on the level of the network.