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

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... plication 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.