  Best match graphs (BMGs) are vertex-colored directed graphs that were
introduced to model the relationships of genes (vertices) from different
species (colors) given an underlying evolutionary tree that is assumed to be
unknown. In real-life applications, BMGs are estimated from sequence similarity
data. Measurement noise and approximation errors usually result in empirically
determined graphs that in general violate characteristic properties of BMGs.
The arc modification problems for BMGs aim at correcting such violations and
thus provide a means to improve the initial estimates of best match data. We
show here that the arc deletion, arc completion and arc editing problems for
BMGs are NP-complete and that they can be formulated and solved as integer
linear programs. To this end, we provide a novel characterization of BMGs in
terms of triples (binary trees on three leaves) and a characterization of BMGs
with two colors in terms of forbidden subgraphs.
