Reconstruction of LGT networks from tri-LGT-nets

Phylogenetic networks have gained attention from the scientific community due to the evidence of the existence of evolutionary events that cannot be represented using trees. A variant of phylogenetic networks, called LGT networks, models specifically lateral gene transfer events, which cannot be properly represented with generic phylogenetic networks. In this paper we treat the problem of the reconstruction of LGT networks from substructures induced by three leaves, which we call tri-LGT-nets. We first restrict ourselves to a class of LGT networks that are both mathematically treatable and biologically significant, called BAN-LGT networks. Then, we study the decomposition of such networks in subnetworks with three leaves and ask whether or not this decomposition determines the network. The answer to this question is negative, but if we further impose time-consistency (species involved in a later gene transfer must coexist) the answer is affirmative, up to some redundancy that can never be recovered but is fully characterized.