Reconstructing Tree-Child Networks from Reticulate-Edge-Deleted Subnetworks

Network reconstruction lies at the heart of phylogenetic research. Two well studied classes of phylogenetic networks include tree-child networks and level-$k$ networks. In a tree-child network, every non-leaf node has a child that is a tree node or a leaf. In a level-$k$ network, the maximum number of reticulations contained in a biconnected component is $k$. Here, we show that level-$k$ tree-child networks are encoded by their reticulate-edge-deleted subnetworks, which are subnetworks obtained by deleting a single reticulation edge, if $k\geq 2$. Following this, we provide a polynomial-time algorithm for uniquely reconstructing such networks from their reticulate-edge-deleted subnetworks. Moreover, we show that this can even be done when considering subnetworks obtained by deleting one reticulation edge from each biconnected component with $k$ reticulations.