Related papers: The maximum agreement subtree problem
The number of embeddings of a partially ordered set $S$ in a partially ordered set $T$ is the number of subposets of $T$ isomorphic to $S$. If both, $S$ and $T$, have only one unique maximal element, we define good embeddings as those in…
Assuming Zipf's Law to be accurate, we show that existing techniques for partially optimizing binary trees produce results that are approximately 10% worse than true optimal. We present a new approximate optimization technique that runs in…
We study multicolour, oriented and high-dimensional discrepancies of the set of all subtrees of a tree. As our main result, we show that the $r$-colour discrepancy of the subtrees of any tree is a linear function of the number of leaves…
The Maximum (Minimum) Leaf Spanning Tree problem asks for a spanning tree with the largest (smallest) number of leaves. As spanning trees are often computed using graph search algorithms, it is natural to restrict this problem to the set of…
We study the distributed detection problem in the context of a balanced binary relay tree, where the leaves of the tree correspond to $N$ identical and independent sensors generating binary messages. The root of the tree is a fusion center…
Tropical geometry with the max-plus algebra has been applied to statistical learning models over tree spaces because geometry with the tropical metric over tree spaces has some nice properties such as convexity in terms of the tropical…
In the longest plane spanning tree problem, we are given a finite planar point set $\mathcal{P}$, and our task is to find a plane (i.e., noncrossing) spanning tree for $\mathcal{P}$ with maximum total Euclidean edge length. Despite more…
We propose the following conjecture: For every fixed $\alpha\in [0,\frac 13)$, each graph of minimum degree at least $(1+\alpha)\frac k2$ and maximum degree at least $2(1-\alpha)k$ contains each tree with $k$ edges as a subgraph. Our main…
Let $\mathcal{T}$ be an unrooted binary tree with $n$ distinctly labelled leaves. Deriving its name from the field of phylogenetics, a convex character on $\mathcal{T}$ is simply a partition of the leaves such that the minimal spanning…
Tree balance plays an important role in various research areas in phylogenetics and computer science. Typically, it is measured with the help of a balance index or imbalance index. There are more than 25 such indices available, recently…
Let G = (V,E) be a finite, simple and undirected graph. For $S \subseteq V$, let $\delta(S,G) = \{(u,v) \in E : u \in S \mbox {and} v \in V-S \}$ be the edge boundary of $S$. Given an integer $i$, $1 \leq i \leq | V |$, let the edge…
Most of major algorithms for phylogenetic tree reconstruction assume that sequences in the analyzed set either do not have any offspring, or that parent sequences can maximally mutate into just two descendants. The graph resulting from such…
A tree ${\mathbb T} =\langle T\leq \rangle$ is reversible iff there is no order $\preccurlyeq \;\varsubsetneq \;\leq $ such that ${\mathbb T} \cong \langle T ,\preccurlyeq\rangle$. Using a characterization of reversibility via back and…
The longest common extension (LCE) of two indices in a string is the length of the longest identical substrings starting at these two indices. The LCE problem asks to preprocess a string into a compact data structure that supports fast LCE…
We estimate the size of a labelled tree by comparing the amount of (labelled) nodes with the size of the set of labels. Roughly speaking, a exponentially big labelled tree, is any labelled tree that has an exponential gap between its size,…
Several indices that measure the degree of balance of a rooted phylogenetic tree have been proposed so far in the literature. In this work we define and study a new index of this kind, which we call the total cophenetic index: the sum, over…
We study the problem of maximizing the number of full degree vertices in a spanning tree $T$ of a graph $G$; that is, the number of vertices whose degree in $T$ equals its degree in $G$. In cubic graphs, this problem is equivalent to…
An out-tree $T$ of a directed graph $D$ is a rooted tree subgraph with all arcs directed outwards from the root. An out-branching is a spanning out-tree. By $l(D)$ and $l_s(D)$ we denote the maximum number of leaves over all out-trees and…
In this paper, we investigate a conjecture by von Haeseler concerning the Maximum Parsimony method for phylogenetic estimation, which was published by the Newton Institute in Cambridge on a list of open phylogenetic problems in 2007. This…
The supertree construction problem is about combining several phylogenetic trees with possibly conflicting information into a single tree that has all the leaves of the source trees as its leaves and the relationships between the leaves are…