Best match graphs arise naturally as the first processing intermediate in algorithms for orthology detection. Let T be a phylogenetic (gene) tree T and [Formula: see text] an assignment of leaves of T to species. The best match graph [Formula: see text] 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 [Formula: see text]. Here, we

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... characterize best match graphs and show that it can be decided in cubic time and quadratic space whether [Formula: see text] derived from a tree in this manner. If the answer is affirmative, there is a unique least resolved tree that explains [Formula: see text], which can also be constructed in cubic time.