  Two genes are xenologs in the sense of Fitch if they are separated by at
least one horizontal gene transfer event. Horizonal gene transfer is asymmetric
in the sense that the transferred copy is distinguished from the one that
remains within the ancestral lineage. Hence xenology is more precisely thought
of as a non-symmetric relation: $y$ is xenologous to $x$ if $y$ has been
horizontally transferred at least once since it diverged from the least common
ancestor of $x$ and $y$. We show that xenology relations are characterized by a
small set of forbidden induced subgraphs on three vertices. Furthermore, each
xenology relation can be derived from a unique least-resolved edge-labeled
phylogenetic tree. We provide a linear-time algorithm for the recognition of
xenology relations and for the construction of its least-resolved edge-labeled
phylogenetic tree. The fact that being a xenology relation is a heritable graph
property, finally has far-reaching consequences on approximation problems
associated with xenology relations.
