Unique solvability of certain hybrid networks from their distances

Phylogenetic relationships among taxa have usually been represented by rooted trees in which the leaves correspond to extant taxa and interior vertices correspond to extinct ancestral taxa. Recently, more general graphs than trees have been investigated in order to be able to represent hybridization, lateral gene transfer, and recombination events. A model is presented in which the genome at a vertex is represented by a binary string. In the presence of hybridization and the absence of convergent evolution and homoplasies, the evolution is modeled by an acyclic digraph. In general, it is shown how distances are computed in terms of the “originating weights” at vertices. An example shows that the distance between two vertices may not correspond to the sum of branch lengths on any path in the graph. If two vertices always have a most recent common ancestor, however, then distances can be measured along certain paths. Sufficient conditions are presented so that all the distances in a network are determined by the distances between leaves, including the root. In particular it is shown how to infer the originating weights at interior vertices from such information.