Characterizing weak compatibility in terms of weighted quartets

In phylogenetics there are various methods available for understanding the evolutionary history of a set of species based on the analysis of its 4-element subsets. Guided by biological data, such techniques usually require the initial computation of a quartet-weight function, i.e., a function that assigns a weight to each bipartition of each 4-element subset into two parts of size two, from which a phylogenetic tree or network is subsequently deduced. It is therefore of interest to characterize quartet-weight functions that correspond precisely to phylogenetic trees or networks. Recently, such characterizations have been presented for phylogenetic trees. Here we provide a 5-point condition for characterizing more general structures called weakly compatible split systems. Such split systems underly the construction of split networks, a special class of phylogenetic networks. This 5-point condition also yields a new characterization of quartet-weight functions that correspond to phylogenetic trees.