Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. AbstractOver the last fifteen years, phylogenetic networks have become a popular tool to analyse relationships between species whose past includes reticulation events such as hybridisation or horizontal gene transfer. However, the space of phylogenetic networks is significantly larger than that of phylogenetic trees, and how to analyse and search this enlarged space remains a poorly understood problem. Inspired by the widely-used rooted subtree prune and regraft (rSPR) operation on rooted phylogenetic trees, we propose a new operation-called subnet prune and regraft (SNPR)-that induces a metric on the space of all rooted phylogenetic networks on a fixed set of leaves. We show that the spaces of several popular classes of rooted phylogenetic networks (e.g. tree child, reticulation visible, and tree based) are connected under SNPR and that connectedness remains for the subclasses of these networks with a fixed number of reticulations. Lastly, we bound the distance between two rooted phylogenetic networks under the SNPR operation, show that it is computationally hard to compute this distance exactly, and analyse how the SNPR-distance between two such networks relates to the rSPR-distance between rooted phylogenetic trees that are embedded in these networks.

Hybridization is an important evolutionary process for many groups of species. Thus, conflicting signals in a data set may not be the result of sampling or modeling errors, but due to the fact that hybridization has played a significant role in the evolutionary history of the species under consideration. Assuming that the initial set of gene trees is correct, a basic problem for biologists is to compute this minimum number of hybridization events to explain this set. In this paper, we describe a new reduction-based algorithm for computing the minimum number, when the initial data set consists of two trees. Although the two-tree problem is NP-hard, our algorithm always gives the exact solution and runs efficiently on many real biological problems. Previous algorithms for the two-tree problem either solve a restricted version of the problem or give an answer with no guarantee of the closeness to the exact solution. We illustrate our algorithm on a grass data set. This new algorithm is freely available for application at either http://www.bi.uni-duesseldorf.de/~linz or http://www.math.canterbury.ac.nz/~cas83.

We show that the problem of computing the hybridization number of two rooted binary phylogenetic trees on the same set of taxa X has a constant factor polynomial-time approximation if and only if the problem of computing a minimum-size feedback vertex set in a directed graph (DFVS) has a constant factor polynomial-time approximation. The latter problem, which asks for a minimum number of vertices to be removed from a directed graph to transform it into a directed acyclic graph, is one of the problems in Karp's seminal 1972 list of 21 NP-complete problems. However, despite considerable attention from the combinatorial optimization community it remains to this day unknown whether a constant factor polynomial-time approximation exists for DFVS. Our result thus places the (in)approximability of hybridization number in a much broader complexity context, and as a consequence we obtain that hybridization number inherits inapproximability results from the problem Vertex Cover. On the positive side, we use results from the DFVS literature to give an O(log r log log r) approximation for hybridization number, where r is the value of an optimal solution to the hybridization number problem.

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