Reconstruction of some hybrid phylogenetic networks with homoplasies from distances

Suppose G is a phylogenetic network given as a rooted acyclic directed graph. Let X be a subset of the vertex set containing the root, all leaves, and all vertices of outdegree 1. A vertex is "regular" if it has a unique parent, and "hybrid" if it has two parents. Consider the case where each gene is binary. Assume an idealized system of inheritance in which no homoplasies occur at regular vertices, but homoplasies can occur at hybrid vertices. Under our model, the distances between taxa are shown to be described using a system of numbers called "originating weights" and "homoplasy weights." Assume that the distances are known between all members of X. Sufficient conditions are given such that the graph G and all the originating and homoplasy weights can be reconstructed from the given distances.