Reconstructing a Phylogenetic Level-1 Network from Quartets

Abstract

We describe a method that will reconstruct an unrooted binary phylogenetic level-1 network on $n$ taxa from the set of all quartets containing a certain fixed taxon, in $O(n^3)$ time. We also present a more general method which can handle more diverse quartet data, but which takes $O(n^6)$ time. Both methods proceed by solving a certain system of linear equations over the two-element field $\mathrm{GF}(2)$. For a general dense quartet set, i.e. a set containing at least one quartet on every four taxa, our $O(n^6)$ algorithm constructs a phylogenetic level-1 network consistent with the quartet set if such a network exists and returns an $O(n^2)$-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.