Reconstructing a phylogenetic level-1 network from quartets

We describe a method that will reconstruct an unrooted binary phylogenetic level-1 network on [Formula: see text] taxa from the set of all quartets containing a certain fixed taxon, in [Formula: see text] time. We also present a more general method which can handle more diverse quartet data, but which takes [Formula: see text] time. Both methods proceed by solving a certain system of linear equations over the two-element field [Formula: see text]. For a general dense quartet set, i.e. a set containing at least one quartet on every four taxa, our [Formula: see text] algorithm constructs a phylogenetic level-1 network consistent with the quartet set if such a network exists and returns an [Formula: see text]-sized certificate of inconsistency otherwise. This answers a question raised by Gambette, Berry and Paul regarding the complexity of reconstructing a level-1 network from a dense quartet set, and more particularly regarding the complexity of constructing a cyclic ordering of taxa consistent with a dense quartet set.