Best match graphs arise naturally as the first processing intermediate in algorithms for orthology detection. Let T be a phylogenetic (gene) tree T and an assignment of leaves of T to species. The best match graph is a digraph that contains an arc from x to y if the genes x and y reside in different species and y is one of possibly many (evolutionary) closest relatives of x compared to all other genes contained in the species . Here, we characterize best match graphs and show that it can be decided in cubic time and quadratic space whether derived from a tree in this manner. If the answer is affirmative, there is a unique least resolved tree that explains , which can also be constructed in cubic time.

Background: Many of the commonly used methods for orthology detection start from mutually most similar pairs of genes (reciprocal best hits) as an approximation for evolutionary most closely related pairs of genes (reciprocal best matches). This approximation of best matches by best hits becomes exact for ultrametric dissimilarities, i.e., under the Molecular Clock Hypothesis. It fails, however, whenever there are large lineage specific rate variations among paralogous genes. In practice, this introduces a high level of noise into the input data for best-hit-based orthology detection methods.Results: If additive distances between genes are known, then evolutionary most closely related pairs can be identified by considering certain quartets of genes provided that in each quartet the outgroup relative to the remaining three genes is known. A priori knowledge of underlying species phylogeny greatly facilitates the identification of the required outgroup. Although the workflow remains a heuristic since the correct outgroup cannot be determined reliably in all cases, simulations with lineage specific biases and rate asymmetries show that nearly perfect results can be achieved. In a realistic setting, where distances data have to be estimated from sequence data and hence are noisy, it is still possible to obtain highly accurate sets of best matches.Conclusion: Improvements of tree-free orthology assessment methods can be expected from a combination of the accurate inference of best matches reported here and recent mathematical advances in the understanding of (reciprocal) best match graphs and orthology relations.Availability: Accompanying software is available at https ://githu b.com/david -schal ler/Asymm eTree .

Two errors in the article Best Match Graphs (Geiß et al. in JMB 78: 2015–2057, 2019) are corrected. One concerns the tacit assumption that digraphs are sink-free, which has to be added as an additional precondition in Lemma 9, Lemma 11, Theorem 4. Correspondingly, Algorithm 2 requires that its input is sink-free. The second correction concerns an additional necessary condition in Theorem 9 required to characterize best match graphs. The amended results simplify the construction of least resolved trees for n-cBMGs, i.e., Algorithm 1. All other results remain unchanged and are correct as stated.