Abstract
Popular methods for exploring the space of rooted phylogenetic trees use rearrangement moves such as rooted Nearest Neighbour Interchange (rNNI) and rooted Subtree Prune and Regraft (rSPR). 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 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.
Highlights
Leaf-labelled trees are routinely used in phylogenetics to depict the relatedness between entities such as species and genes
Popular methods for exploring the space of rooted phylogenetic trees use rearrangement moves such as rooted Nearest Neighbour Interchange and rooted Subtree Prune and Regraft. 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
Several tree rearrangement moves have been defined in the past, the most commonly used ones being Nearest Neighbour Interchange (NNI) moves and Subtree Prune and Regraft (SPR) moves; when the phylogenetic trees are considered as rooted, i.e. directed and out-branching, we have their rooted versions: rNNI and rSPR moves
Summary
Leaf-labelled trees are routinely used in phylogenetics to depict the relatedness between entities such as species and genes. Note that a handful of rearrangement heuristics for phylogenetic networks have been published recently (see, for example, PhyloNET (Than et al 2008) and the BEAST 2 add-on SpeciesNetwork) and each of them uses its own set of rearrangement moves, e.g. the “Branch relocator” move in (Zhang et al 2017) and the “Relocating the source of an edge” and “Relocating the destination of a reticulation edge” moves in (Yu et al 2014) These sets of moves are often included among the ones cited in the previous paragraph; for example, the above-cited moves from (Yu et al 2014) correspond, respectively, to head and tail moves (see Definitions 2.5 and 2.6) and their union correspond to the rSPR moves defined in (Gambette et al 2017), c.f. section. We show that the computation of a tail move sequence or rSPR sequence of shortest length is NP-hard
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