  Quasi-best match graphs (qBMGs) are a hereditary class of directed, properly
vertex-colored graphs. They arise naturally in mathematical phylogenetics as a
generalization of best match graphs, which formalize the notion of evolutionary
closest relatedness of genes (vertices) in multiple species (vertex colors).
They are explained by rooted trees whose leaves correspond to vertices. In
contrast to BMGs, qBMGs represent only best matches at a restricted
phylogenetic distance. We provide characterizations of qBMGs that give rise to
polynomial-time recognition algorithms and identify the BMGs as the qBMGs that
are color-sink-free. Furthermore, two-colored qBMGs are characterized as
directed graphs satisfying three simple local conditions, two of which have
appeared previously, namely bi-transitivity in the sense of Das et al. (2021)
and a hierarchy-like structure of out-neighborhoods, i.e., $N(x)\cap
N(y)\in\{N(x),N(y),\emptyset\}$ for any two vertices $x$ and $y$. Further
results characterize qBMGs that can be explained by binary phylogenetic trees.
