Abstract. The graph-theoretic operation of rooted subtree prune and regraft is increasingly being used as a tool for understanding and modelling reticulation events in evolutionary biology. In this paper, we show that computing the rooted subtree prune and regraft distance between two rooted binary phylogenetic trees on the same label set is NP-hard. This resolves a longstanding open problem. Furthermore, we show that this distance is fixed parameter tractable when parameterised by the distance between the two trees.

Acyclic directed graphs (ADGs) are increasingly being viewed as more appropriate for representing certain evolutionary relationships, particularly in biology, than rooted trees. In this paper, we develop a framework for the analysis of these graphs which we call hybrid phylogenies. We are particularly interested in the problem whereby one is given a set of phylogenetic trees and wishes to determine a hybrid phylogeny that 'embeds' each of these trees and which requires the smallest number of hybridisation events. We show that this quantity can be greatly reduced if additional species are involved, and investigate other combinatorial aspects of this and related questions.

Abstract. Evolutionary processes such as hybridisation, lateral gene transfer, and recombination are all key factors in shaping the structure of genes and genomes. However, since such processes are not always best represented by trees, there is now considerable interest in using more general networks instead. For example, in recent studies it has been shown that networks can be used to provide lower bounds on the number of recombination events and also for the number of lateral gene transfers that took place in the evolutionary history of a set of molecular sequences. In this paper we describe the theoretical performance of some related bounds that result when merging pairs of trees into networks.

It is now well-documented that the structure of evolutionary relationships between a set of present-day species is not necessarily tree-like. The reason for this is that reticulation events such as hybridizations mean that species are a mixture of genes from different ancestors. Since such events are relatively rare, a fundamental problem for biologists is to determine the smallest number of hybridization events required to explain a given (input) set of data in a single (hybrid) phylogeny. The main results of this paper show that computing this smallest number is APX-hard, and thus NP-hard, in the case the input is a collection of phylogenetic trees on sets of present-day species. This answers a problem which was raised at a recent conference (Phylogenetic Combinatorics and Applications, Uppsala University, 2004). As a consequence of these results, we also correct a previously published NP-hardness proof in the case the input is a collection of binary sequences, where each sequence represents the attributes of a particular present-day species. The APX-hardness of these problems means that it is unlikely that there is an efficient algorithm for either computing the result exactly or approximating it to any arbitrary degree of accuracy.

The amalgamation of leaf-labelled (phylogenetic) trees on overlapping leaf sets into one (super)tree is a central problem in several areas of classification, particularly evolutionary biology. In this paper, we describe a new technique for amalgamating rooted phylogenetic trees. This appears to be the first such method to provably exhibit particular desirable properties which we list and establish.

Tutte defined a k-separation of a matroid M to be a partition ðA; BÞ of the ground set of M such that jAj; jBjXk and rðAÞ þ rðBÞ À rðMÞok: If, for all mon; the matroid M has no mseparations, then M is n-connected. Earlier, Whitney showed that ðA; BÞ is a 1-separation of M if and only if A is a union of 2-connected components of M: When M is 2-connected, Cunningham and Edmonds gave a tree decomposition of M that displays all of its 2-separations. When M is 3-connected, this paper describes a tree decomposition of M that displays, up to a certain natural equivalence, all non-trivial 3-separations of M: r

We describe some new and recent results that allow for the analysis and representation of reticulate evolution by non-tree networks. In particular, we (1) present a simple result to show that, despite the presence of reticulation, there is always a well-defined underlying tree that corresponds to those parts of life that do not have a history of reticulation; (2) describe and apply new theory for determining the smallest number of hybridization events required to explain conflicting gene trees; and (3) present a new algorithm to determine whether an arbitrary rooted network can be realized by contemporaneous reticulation events. We illustrate these results with examples. [Directed acyclic graph; reticulate evolution; hybrid species; sub-tree prune and re-graft.].

. (2007) 'Computing the hybridisation number of two phylogenetic trees is xed parameter tractable.', Transactions on computational biology and bioinformatics., 4 (3). pp. 458-466. Further information on publisher's website:http://dx.doi.org/10. 1109/tcbb.2007.1019 Publisher's copyright statement: c 2007 IEEE. This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder Personal use of this material is permitted.However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE. Additional information: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. Abstract-Reticulation processes in evolution mean that the ancestral history of certain groups of present-day species is non-tree-like. These processes include hybridization, lateral gene transfer, and recombination. Despite the existence of reticulation, such events are relatively rare and, so, a fundamental problem for biologists is the following: Given a collection of rooted binary phylogenetic trees on sets of species that correctly represent the tree-like evolution of different parts of their genomes, what is the smallest number of "reticulation" vertices in any network that explains the evolution of the species under consideration? It has been previously shown that this problem is NP-hard even when the collection consists of only two rooted binary phylogenetic trees. However, in this paper, we show that the problem is fixed-parameter tractable in the two-tree instance when parameterized by this smallest number of reticulation vertices.Index Terms-Rooted phylogenetic tree, reticulate evolution, hybridization network, agreement forest, subtree prune and regraft.