Before revision:  We consider duplication/loss scenarios and characterize unambiguous false-positive (u-fp) orthology assignments, that is, edges in the Best Match Graphs (BMGs) that cannot correspond to orthologs for any gene tree that explains the BMG. We characterize u-fp edges in terms of subgraphs of the BMG and show that, given a BMG, there is a unique "augmented tree" that explains the BMG and identifies all u-fp edges in terms of overlapping sets of species in certain subtrees. The augmented tree can be constructed as a refinement of the unique least resolved tree of the BMGin polynomial time. Removal of the u-fp edges from the reciprocal best matches results in a unique orthology assignment.
After revision:  We consider duplication/loss scenarios and characterize unambiguous false-positive (u-fp) orthology assignments, that is, edges in the Best Match Graphs (BMGs) that cannot correspond to orthologs for any gene tree that explains the BMG. We characterize u-fp edges in terms of subgraphs of the BMG and show that, given a BMG, there is a unique "augmented tree" that explains the BMG and identifies all u-fp orthology assignments in a BMG. Simulations show that at least 75