  Orthology and paralogy relations are often inferred by methods based on gene
similarity, which usually yield a graph depicting the relationships between
gene pairs. Such relation graphs are known to frequently contain errors, as
they cannot be explained via a gene tree that both contains the depicted
orthologs/paralogs, and that is consistent with a species tree $S$. This idea
of detecting errors through inconsistency with a species tree has mostly been
studied in the presence of speciation and duplication events only. In this
work, we ask: could the given set of relations be consistent if we allow
lateral gene transfers in the evolutionary model? We formalize this question
and provide a variety of algorithmic results regarding the underlying problems.
Namely, we show that deciding if a relation graph $R$ is consistent with a
given species network $N$ is NP-hard, and that it is W[1]-hard under the
parameter "minimum number of transfers". However, we present an FPT algorithm
based on the degree of the $DS$-tree associated with $R$. We also study
analogous problems in the case that the transfer highways on a species tree are
unknown.
