Quarnet inference rules for level-1 networks

An important problem in phylogenetics is the construction of phylogenetic trees. One way to approach this problem, known as the supertree method, involves inferring a phylogenetic tree with leaves consisting of a set $X$ of species from a collection of trees, each having leaf-set some subset of $X$. In the 1980's characterizations, certain inference rules were given for when a collection of 4-leaved trees, one for each 4-element subset of $X$, can all be simultaneously displayed by a single supertree with leaf-set $X$. Recently, it has become of interest to extend such results to phylogenetic networks. These are a generalization of phylogenetic trees which can be used to represent reticulate evolution (where species can come together to form a new species). It has been shown that a certain type of phylogenetic network, called a level-1 network, can essentially be constructed from 4-leaved trees. However, the problem of providing appropriate inference rules for such networks remains unresolved. Here we show that by considering 4-leaved networks, called quarnets, as opposed to 4-leaved trees, it is possible to provide such rules. In particular, we show that these rules can be used to characterize when a collection of quarnets, one for each 4-element subset of $X$, can all be simultaneously displayed by a level-1 network with leaf-set $X$. The rules are an intriguing mixture of tree inference rules, and an inference rule for building up a cyclic ordering of $X$ from orderings on subsets of $X$ of size 4. This opens up several new directions of research for inferring phylogenetic networks from smaller ones, which could yield new algorithms for solving the supernetwork problem in phylogenetics.