Exploring the tiers of rooted phylogenetic network space using tail moves

Popular methods for exploring the space of rooted phylogenetic trees use rearrangement moves such as rNNI (rooted Nearest Neighbour Interchange) and rSPR (rooted Subtree Prune and Regraft). Recently, these moves were generalized to rooted phylogenetic networks, which are a more suitable representation of reticulate evolutionary histories, and it was shown that any two rooted phylogenetic networks of the same complexity are connected by a sequence of either rSPR or rNNI moves. Here, we show that this is possible using only tail moves, which are a restricted version of rSPR moves on networks that are more closely related to rSPR moves on trees. The connectedness still holds even when we restrict to distance-1 tail moves (a localized version Remie Janssen Delft Institute of Applied Mathematics, Delft University of Technology Postbus 5031,2600 GA Delft, The Netherlands E-mail: [email protected] Mark Jones Delft Institute of Applied Mathematics, Delft University of Technology Postbus 5031,2600 GA Delft, The Netherlands E-mail: [email protected] Péter Erdős MTA Rényi Institute of Mathematics, Reáltanoda u 13-15 Budapest, 1053 Hungary E-mail: [email protected] Corresponding author: L. van Iersel Delft Institute of Applied Mathematics, Delft University of Technology Postbus 5031,2600 GA Delft, The Netherlands E-mail: [email protected] C. Scornavacca Institut des Sciences de l’Evolution, Université de Montpellier, CNRS, IRD, EPHE Institut de Biologie Computationnelle (IBC) Place Eugène Bataillon, Montpellier, France E-mail: [email protected] 2 Remie Janssen et al. of tail-moves). Moreover, we give bounds on the number of (distance-1) tail moves necessary to turn one network into another, which in turn yield new bounds for rSPR, rNNI and SPR (i.e. the equivalent of rSPR on unrooted networks). The upper bounds are constructive, meaning that we can actually find a sequence with at most this length for any pair of networks. Finally, we show that finding a shortest sequence of tail or rSPR moves is NP-hard. Mathematics Subject Classification (2000) 92D15 · 68R10 · 68R05 · 05C20