Analyzing and reconstructing reticulation networks under timing constraints

Reticulation networks are now frequently used to model the history of life for various groups of organisms whose evolutionary past is likely to include reticulation events like horizontal gene transfer or hybridization. However, the reconstructed networks are rarely guaranteed to be temporal. If a reticulation network is temporal, then it satisfies the two biologically motivated timing constraints of instantaneously occurring reticulation events and successively occurring speciation events. On the other hand, if a reticulation network is not temporal, it is always possible to resolve this issue by adding a number of additional unsampled or extinct taxa. In the first half of the paper, we show that deciding whether a given number of additional taxa is sufficient to transform a non-temporal reticulation network into a temporal one is an NP-complete problem. As one is often given a set of gene trees instead of a network in the context of hybridization, this motivates the second half of the paper which provides an algorithm for reconstructing a temporal hybridization network that simultaneously explains the ancestral history of two trees or indicates that no such network exists. We highlight two practical applications of this algorithm and illustrate the second application on a grass data set