  Introgression is an evolutionary process in which genes or other types of
genetic material are introduced into a genome. It is an important evolutionary
process that can, for example, play a fundamental role in speciation. Recently
the concept of an overlaid species forest was introduced to represent
introgression histories. Basically this approach takes a putative gene history
in the form of a phylogenetic gene tree and tries to overlay this onto a forest
which usually consists of a collection of lineage trees for the species of
interest. The result is a network called an overlaid species forest in which
genes jump or introgress between lineages. In this paper we study properties of
overlaid species forests, showing that they have various connections with
models for lateral gene transfer, maximum parsimony, and unfolding of
phylogenetic networks. In particular, we show that a certain algorithm called
OSF-B UILDER for constructing overlaid species forests is guaranteed to a
produce a special type of overlaid species forest with a minimum number
introgressions, as well as providing some characterizations for networks that
can arise from overlaid species forests. We expect that these results will be
useful in developing new methods for representing introgression histories, a
growing area of interest in phylogenetics.
