Defining Phylogenetic Network Distances Using Cherry Operations

In phylogenetic networks, picking a cherry consists of removing a leaf that shares a parent with another leaf, or removing a reticulate edge whose endpoints are parents of leaves. Cherry-picking operations were recently shown to have several structural and algorithmic applications in the study of networks, for instance in determining their reconstructibility or in solving the network hybridization and network containment problems. In particular, some networks within certain classes are isomorphic if they can be reduced to a single node by the same sequence of cherry-picking operations. Therefore, cherry-picking sequences contain information on the level of similarity between two networks. In this paper, we expand on this idea by devising four novel distances on networks based on cherry picking and their reverse operation. We provide bounds between these distances and show that three of them are equal despite their different formulations. We also show that computing these three equivalent distances is NP-hard, even when restricted to comparing a tree and a network. On the positive side, we show that they can be computed in quadratic time on two trees, providing a new comparative measure for phylogenetic trees that can be computed efficiently.