The structure of equivalent 3-separations in a 3-connected matroid

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The structure of 3-connected matroids of path width three

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Reconstructing minimal rooted trees

For a set T of rooted binary leaf-labelled trees, we present an algorithm that finds all of the minor-minimal trees that are compatible with T . The running time of this algorithm is polynomial up to the number of trees with this property. This type of problem arises in several areas of classifcation, particularly evolutionary biology

Fork-decompositions of matroids

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Reconstructing minimal rooted trees

For a set T of rooted binary leaf-labelled trees, we present an algorithm that finds all of the minor-minimal trees that are compatible with T . The running time of this algorithm is polynomial up to the number of trees with this property. This type of problem arises in several areas of classifcation, particularly evolutionary biology

On matroids of branch-width three

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On the computational complexity of the rooted subtree prune and regraft distance

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 xed parameter tractable when parameterised by the distance between the two trees

Fast Computation of Supertrees for Compatible Phylogenies with Nested Taxa

Typically, supertree methods combine a collection of source trees in which just the leaves are labeled by taxa. In such methods the resulting supertree is also leaf labeled. An underlying assumption in these methods is that across all trees in the collection, no two of the taxa are nested; for example, "buttercups" and "plants" are nested taxa. Motivated by Page, the first supertree algorithm for allowing the source trees to collectively have nested taxa is called AncestralBuild. Here, in addition to taxa labeling the leaves, the source trees may have taxa labeling some of their interior nodes. Taxa-labeling interior nodes are at a higher taxonomic level than that of their descendants (for example, genera versus species). Analogous to the supertree method Build for deciding the compatibility of a collection of source trees in which just the leaves are labeled, AncestralBuild is a polynomial-time algorithm for deciding the compatibility of a collection of source trees in which some of the interior nodes are also labeled by taxa. Although a more general method, in this paper we show that the original description of AncestralBuild can be modified so that the running time is as fast as the current fastest running time for Build. Fast computation for deciding compatibility is essential if one is to make use of phylogenetic databases that contain thousands of trees on tens of thousands of taxa. This is particularly so as AncestralBuild is incorporated as a basic tool inside more general supertree methods (that is, methods that always output a tree regardless of the compatibility of the source trees). We apply the method to propose a comprehensive phylogeny of the strepsirrhines, a major group of the primates

Determining phylogenetic networks from inter-taxa distances

We consider the problem of determining the topological structure of a phylogenetic network given only information about the path-length distances between taxa. In particular, one of the main results of the paper shows that binary tree-child networks are essentially determined by such information