Unique determination of some homoplasies at hybridization events

Phylogenetic relationships may be represented by rooted acyclic directed graphs in which each vertex, corresponding to a taxon, possesses a genome. Assume the characters are all binary. A homoplasy occurs if a particular character changes its state more than once in the graph. A vertex is "regular" if it has only one parent and "hybrid" if it has more than one parent. A "regular path" is a directed path such that all vertices after the first are regular. Assume that the network is given and that the genomes are known for all leaves and for the root. Assume that all homoplasies occur only at hybrid vertices and each character has at most one homoplasy. Assume that from each vertex there is a regular path leading to a leaf. In this idealized setting, with other mild assumptions, it is proved that the genome at each vertex is uniquely determined. Hence, for each character the vertex at which a homoplasy occurs in the character is uniquely determined. Without the assumption on regular paths, an example shows that the genomes and homoplasies need not be uniquely determined.