Combining Orthology and Xenology Data in a Common Phylogenetic Tree

In mathematical phylogenetics, types of events in a gene tree T are formalized by vertex labels t(v) and set-valued edge \(\lambda (e)\). The orthology paralogy relations between genes special case map \(\delta \) on the pairs leaves defined (x,y)=q\) if last common ancestor \({\text {lca}}(x,y)\) x y is labeled an event type q, e.g., speciation or duplication. Similarly, \(\varepsilon with \(m\in \varepsilon (x,y)\) \lambda (e)\) for at least one e along path from to generalizes xenology, i.e., horizontal transfer. We show that pair maps \((\delta ,\varepsilon )\) derives \((T,t,\lambda this manner only there exists refinement (unique) least-resolved \((T_{\delta },t_{\delta })\) explains \((T_{\varepsilon },\lambda _{\varepsilon (provided both trees exist). This result remains true certain combinations incident vertices edges forbidden.