Molecular data sets characterized by few phylogenetically informative characters with a broad spectrum of mutation rates, such as intraspecific controlregion sequence variation of human mitochondrial DNA (mtDNA), can be usefully visualized in the form of median networks. Here we provide a step-by-step guide to the construction of such networks by hand. We improve upon a previously implemented algorithm by outlining an efficient parametrized strategy amenable to large data sets, greedy reduction, which makes it possible to reconstruct some of the confounding recurrent mutations. This entails some postprocessing as well, which assists in capturing more parsimonious solutions. To simplify the creation of the resulting network by hand, we describe a speedy approach to network construction, based on a careful planning of the processing order. A coalescent simulation tailored to human mtDNA variation in Eurasia testifies to the usefulness of reduced median networks, while highlighting notorious problems faced by all phylogenetic methods in this context. Finally, we discuss two case studies involving the comparison of characters in the two hypervariable segments of the human mtDNA control region in the light of the worldwide control-region sequence database, as well as additional restriction fragment length polymorphism information. We conclude that only a minority of the mutations that hit the second segment occur at sites that would have a mutation rate comparable to those at most sites in the first segment. Discarding the known “noisy” sites of the second segment enhances the analysis.

We introduce a new phylogenetic reconstruction algorithm which, unlike most previous rigorous inference techniques, does not rely on assumptions regarding the branch lengths or the depth of the tree. The algorithm returns a forest which is guaranteed to contain all edges that are: 1) sufficiently long and 2) sufficiently close to the leaves. How much of the true tree is recovered depends on the sequence length provided. The algorithm is distance-based and runs in polynomial time. 1

A graph model for a set S of splits of a set X consists of a graph and a map from X to the vertices of the graph such that the inclusion-minimal cuts of the graph represent S. Phylogenetic trees are graph models in which the graph is a tree. We show that the model can be generalized to a cactus (i.e. a tree of edges and cycles) without losing computational efficiency. A cactus can represent a quadratic rather than linear number of splits in linear space. We show how to decide in time linear in the size of a succinct representation of S whether a set of splits has a cactus model, and if so construct it within the same time bounds. As a byproduct, we show how to construct the subset of all compatible splits and a maximal compatible set of splits in linear time. Note that it is N P-complete to find a compatible subset of maximum size. Finally, we briefly discuss further generalizations of tree models.