Related papers: Minimum triplet covers of binary phylogenetic $X$-…
Tanglegrams are drawings of two rooted binary phylogenetic trees and a matching between their leaf sets. The trees are drawn crossing-free on opposite sides with their leaf sets facing each other on two vertical lines. Instead of minimizing…
Given a digraph $D$, the Minimum Leaf Out-Branching problem (MinLOB) is the problem of finding in $D$ an out-branching with the minimum possible number of leaves, i.e., vertices of out-degree 0. We prove that MinLOB is polynomial-time…
Balanced minimum evolution is a distance-based criterion for the reconstruction of phylogenetic trees. Several algorithms exist to find the optimal tree with respect to this criterion. One approach is to minimize a certain linear functional…
Distances on merge trees facilitate visual comparison of collections of scalar fields. Two desirable properties for these distances to exhibit are 1) the ability to discern between scalar fields which other, less complex topological…
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…
The Rooted Maximum Leaf Outbranching problem consists in finding a spanning directed tree rooted at some prescribed vertex of a digraph with the maximum number of leaves. Its parameterized version asks if there exists such a tree with at…
We discuss a category of graphs, recursive clique trees, which have small-world and scale-free properties and allow a fine tuning of the clustering and the power-law exponent of their discrete degree distribution. We determine relevant…
We address the problem of finding nice labellings for event structures of degree 3. We develop a minimum theory by which we prove that the labelling number of an event structure of degree 3 is bounded by a linear function of the…
In the critical beta-splitting model of a random $n$-leaf rooted tree, clades are recursively split into sub-clades, and a clade of $m$ leaves is split into sub-clades containing $i$ and $m-i$ leaves with probabilities $\propto 1/(i(m-i))$.…
We show that for any two values $\alpha, \beta >0 $ for which $\alpha+\beta>1$ then there is a value $N$ so that for all $n \geq N$ the following holds. For any binary phylogenetic tree $T$ on $n$ leaves there is a set of $\lfloor n^\alpha…
The problem of characterizing trees with minimal atom-bond-connectivity index (minimal-ABC trees) has a reputation as one of the most demanding recent open optimization problems in mathematical chemistry. Here firstly, we give an…
We characterize classes of graphs closed under taking vertex-minors and having no $P_n$ and no disjoint union of $n$ copies of the $1$-subdivision of $K_{1,n}$ for some $n$. Our characterization is described in terms of a tree of radius $2$…
A tanglegram consists of two binary rooted trees with the same number of leaves and a perfect matching between the leaves of the trees. We show that the two halves of a random tanglegram essentially look like two independently chosen random…
We consider packing tree degree sequences in this paper. We set up a conjecture that any arbitrary number of tree degree sequences without common leaves have edge disjoint tree realizations. This conjecture is known to be true for $2$ and…
Merge trees, a type of topological descriptor, serve to identify and summarize the topological characteristics associated with scalar fields. They present a great potential for the analysis and visualization of time-varying data. First,…
Three-way dissimilarities are a generalization of (two-way) dissimilarities which can be used to indicate the lack of homogeneity or resemblance between any three objects. Such maps have applications in cluster analysis, and have been used…
In this paper we investigate an extremal problem on binary phylogenetic trees. Given two such trees $T_1$ and $T_2$, both with leaf-set ${1,2,...,n}$, we are interested in the size of the largest subset $S \subseteq {1,2,...,n}$ of leaves…
Measures of tree balance play an important role in the analysis of phylogenetic trees. One of the oldest and most popular indices in this regard is the Colless index for rooted bifurcating trees, introduced by Colless (1982). While many of…
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 phylogenetic tree is an edge-weighted binary tree, with leaves labelled by a collection of species, that represents the evolutionary relationships between those species. For such a tree, a phylogenetic diversity index is a function that…