  2-colored best match graphs (2-BMGs) form a subclass of sink-free
bi-transitive graphs that appears in phylogenetic combinatorics. There, 2-BMGs
describe evolutionarily most closely related genes between a pair of species.
They are explained by a unique least resolved tree (LRT). Introducing the
concept of support vertices we derive an $O(|V|+|E|\log^2|V|)$-time algorithm
to recognize 2-BMGs and to construct its LRT. The approach can be extended to
also recognize binary-explainable 2-BMGs with the same complexity. An empirical
comparison emphasizes the efficiency of the new algorithm.
