Related papers: On Finding Directed Trees with Many Leaves
Graph searches and their respective search trees are widely used in algorithmic graph theory. The problem whether a given spanning tree can be a graph search tree has been considered for different searches, graph classes and search tree…
The Minimum Branch Vertices Spanning Tree problem aims to find a spanning tree $T$ in a given graph $G$ with the fewest branch vertices, defined as vertices with a degree three or more in $T$. This problem, known to be NP-hard, has…
Phylogenetic networks generalize phylogenetic trees by allowing the modelization of events of reticulate evolution. Among the different kinds of phylogenetic networks that have been proposed in the literature, the subclass of binary…
We develop two different methods to achieve subexponential time parameterized algorithms for problems on sparse directed graphs. We exemplify our approaches with two well studied problems. For the first problem, {\sc $k$-Leaf…
A new tree model is introduced based on ordered trees, by distinguishing exactly one child of each node that \emph{has} children. The basic enumeration leads to a cubic equation of the generating function. The extraction of its coefficients…
A linear forest is a collection of vertex-disjoint paths. The Linear Arboricity Conjecture states that every graph of maximum degree $\Delta$ can be decomposed into at most $\lceil(\Delta+1)/2\rceil$ linear forests. We prove that $\Delta/2…
Decision Tree (DT) Learning is a fundamental problem in Interpretable Machine Learning, yet it poses a formidable optimisation challenge. Practical algorithms have recently emerged, primarily leveraging Dynamic Programming and Branch &…
This paper studies two classes of variational problems introduced in [7], related to the optimal shapes of tree roots and branches. Given a measure $\mu$ describing the distribution of leaves, a sunlight functional $\S(\mu)$ computes the…
Measures of tree balance play an important role in different research areas such as mathematical phylogenetics or theoretical computer science. The balance of a tree is usually quantified in a single number, called a balance or imbalance…
An algorithm is proposed for constructing directed spanning forests of the minimum weight, in which the maximum possible degree of affinity between the minimum forests is preserved when the number of trees changes. The correctness of the…
We present recursive formulas giving the maximal number of leaves in tree-like polyforms living in two-dimensional regular lattices and in tree-like polycubes in the three-dimensional cubic lattice. We call these tree-like polyforms and…
In the present paper we consider the problem of constructing all the projective rooted spanning trees of a given graph. We propose an algorithm based on reducing this problem to the problem of constructing all the maximal independent sets…
We consider the problem of embedding the Steiner points of a Steiner tree with given topology into the rectilinear plane. Thereby, the length of the path between a distinguished terminal and each other terminal must not exceed given length…
Understanding the evolution of a set of genes or species is a fundamental problem in evolutionary biology. The problem we study here takes as input a set of trees describing {possibly discordant} evolutionary scenarios for a given set of…
We consider two problems for a directed graph $G$, which we show to be closely related. The first one is to find $k$ edge-disjoint forests in $G$ of maximal size such that the indegree of each vertex in these forests is at most $k$. We…
In this paper we consider the problem of connected edge searching of weighted trees. It is shown that there exists a polynomial-time algorithm for finding optimal connected search strategy for bounded degree trees with arbitrary weights on…
A variant of the Erd\H{o}s-S\'os conjecture, posed by Havet, Reed, Stein and Wood, states that every graph with minimum degree at least $\lfloor 2k/3 \rfloor$ and maximum degree at least $k$ contains a copy of every tree with $k$ edges.…
We study the problem of maximizing the number of spanning trees in a connected graph by adding at most $k$ edges from a given candidate edge set. We give both algorithmic and hardness results for this problem: - We give a greedy algorithm…
We investigate the tractability of a simple fusion of two fundamental structures on graphs, a spanning tree and a perfect matching. Specifically, we consider the following problem: given an edge-weighted graph, find a minimum-weight…
We prove the meridional rank conjecture for arborescent links associated to plane trees with the following property: all branching points carry a straight branch to at least three leaves. The proof involves an upper bound on the bridge…