Title: Quarnet inference rules for level-1 networks,
Abstract: 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.