Related papers: A matroid associated with a phylogenetic tree
A classical result, fundamental to evolutionary biology, states that an edge-weighted tree $T$ with leaf set $X$, positive edge weights, and no vertices of degree 2 can be uniquely reconstructed from the set of leaf-to-leaf distances…
The tree-metric theorem provides a necessary and sufficient condition for a dissimilarity matrix to be a tree metric, and has served as the foundation for numerous distance-based reconstruction methods in phylogenetics. Our main result is…
A classical result in distance based tree-reconstruction characterizes when for a distance $D$ on some finite set $X$ there exist a uniquely determined dendrogram on $X$ (essentially a rooted tree $T=(V,E)$ with leaf set $X$ and no degree…
A metric phylogenetic tree relating a collection of taxa induces weighted rooted triples and weighted quartets for all subsets of three and four taxa, respectively. New intertaxon distances are defined that can be calculated from these…
The construction of a dendogram on a set of individuals is a key component of a genomewide association study. However even with modern sequencing technologies the distances on the individuals required for the construction of such a…
Phylogenetic trees are a central tool in understanding evolution. They are typically inferred from sequence data, and capture evolutionary relationships through time. It is essential to be able to compare trees from different data sources…
Tree-based networks are a class of phylogenetic networks that attempt to formally capture what is meant by "tree-like" evolution. A given non-tree-based phylogenetic network, however, might appear to be very close to being tree-based, or…
Ultametrics are an important class of distances used in applications such as phylogenetics, clustering and classification theory. Ultrametrics are essentially distances that can be represented by an edge-weighted rooted tree so that all of…
It is known that there is an alternative characterization of characteristic vertices for trees with positive weights on their edges via Perron values and Perron branches. Moreover, the algebraic connectivity of a tree with positive edge…
Phylogenetic networks which are, as opposed to trees, suitable to describe processes like hybridization and horizontal gene transfer, play a substantial role in evolutionary research. However, while non-treelike events need to be taken into…
We study distorted metrics on binary trees in the context of phylogenetic reconstruction. Given a binary tree $T$ on $n$ leaves with a path metric $d$, consider the pairwise distances $\{d(u,v)\}$ between leaves. It is well known that these…
Phylogenetic networks are a generalization of phylogenetic trees that allow for representation of reticulate evolution. Recently, a space of unrooted phylogenetic networks was introduced, where such a network is a connected graph in which…
Dissimilarity measures for (possibly weighted) phylogenetic trees based on the comparison of their vectors of path lengths between pairs of taxa, have been present in the systematics literature since the early seventies. But, as far as…
As an extension of a classical tree-partition problem, we consider decompositions of graphs into edge-disjoint (rooted-)trees with an additional matroid constraint. Specifically, suppose we are given a graph $G=(V,E)$, a multiset…
Motivated by applications to low-rank matrix completion, we give a combinatorial characterization of the independent sets in the algebraic matroid associated to the collection of $m\times n$ rank-2 matrices and $n\times n$ skew-symmetric…
The early development of a zygote can be mathematically described by a developmental tree. To compare developmental trees of different species, we need to define distances on trees. If children cells after a division are not…
A phylogenetic tree shows the evolutionary relationships among species. Internal nodes of the tree represent speciation events and leaf nodes correspond to species. A goal of phylogenetics is to combine such trees into larger trees, called…
Generalizing a well known theorem for finite matroids, we prove that for every (infinite) connected matroid M there is a unique tree T such that the nodes of T correspond to minors of M that are either 3-connected or circuits or cocircuits,…
We study the conditions under which the isometry of spaces with metrics generated by weights given on the edges of finite trees is equivalent to the isomorphism of these trees. Similar questions are studied for ultrametric spaces generated…
We attempt to shed new light on the notion of 'tree-like' metric spaces by focusing on an approach that does not use the four-point condition. Our key question is: Given metric space $M$ on $n$ points, when does a fully labelled…