Related papers: A Computational Approach to the Graceful Tree Conj…
We propose a new methodology to develop heuristic algorithms using tree decompositions. Traditionally, such algorithms construct an optimal solution of the given problem instance through a dynamic programming approach. We modify this…
The linear arboricity la(G) of a graph G is the minimum number of linear forests that partition the edges of G. In 1984, Akiyama et al. stated the Linear Arboricity Conjecture (LAC), that the linear arboricity of any simple graph of maximum…
Over some types of trees with a given number of vertices, which trees minimize or maximize the total number of subtrees or leaf containing subtrees are studied. Here are some of the main results:\ (1)\, Sharp upper bound on the total number…
Motivated by alignment of correlated sparse random graphs, we introduce a hypothesis testing problem of deciding whether or not two random trees are correlated. We obtain sufficient conditions under which this testing is impossible or…
The Strong Nine Dragon Tree Conjecture asserts that for any integers $k$ and $d$ any graph with fractional arboricity at most $k + \frac{d}{d+k+1}$ decomposes into $k+1$ forests, such that for at least one of the forests, every connected…
The transmission of a vertex in a connected graph is the sum of distances from that vertex to all the other vertices. A connected graph is transmission irregular if any two distinct vertices have different transmissions. We present an…
Tree Containment is a fundamental problem in phylogenetics useful for verifying a proposed phylogenetic network, representing the evolutionary history of certain species. Tree Containment asks whether the given phylogenetic tree (for…
The implicit graph conjecture states that every sufficiently small, hereditary graph class has a labeling scheme with a polynomial-time computable label decoder. We approach this conjecture by investigating classes of label decoders defined…
Mechanical exfoliation of graphene and its identification by optical inspection is one of the milestones in condensed matter physics that sparked the field of 2D materials. Finding regions of interest from the entire sample space and…
A function $f$ is a \textit{graceful labelling} of a graph $G=(V,E)$ with $m$ edges if $f$ is an injection $f:V\mapsto \{0,1,2,\dots,m\}$ such that each edge $uv \in E$ is assigned the label $|f(u)-f(v)|$, and no two edge labels are the…
We prove, that every connected graph with $s$ vertices of degree 3 and $t$ vertices of degree at least~4 has a spanning tree with at least ${2\over 5}t +{1\over 5}s+\alpha$ leaves, where $\alpha \ge {8\over 5}$. Moreover, $\alpha \ge 2$ for…
Let $T$ be a distinguished subset of vertices in a graph $G$. A $T$-\emph{Steiner tree} is a subgraph of $G$ that is a tree and that spans $T$. Kriesell conjectured that $G$ contains $k$ pairwise edge-disjoint $T$-Steiner trees provided…
A "tree-partition" of a graph $G$ is a partition of $V(G)$ such that identifying the vertices in each part gives a tree. It is known that every graph with treewidth $k$ and maximum degree $\Delta$ has a tree-partition with parts of size…
Decision trees have been a very popular class of predictive models for decades due to their interpretability and good performance on categorical features. However, they are not always robust and tend to overfit the data. Additionally, if…
We find surprisingly simple formulas for the limiting probability that the rank of a randomly selected vertex in a randomly selected phylogenetic tree or generalized phylogenetic tree is a given integer.
In this paper we describe an algorithm visiting all numerical semigroups up to a given genus using a well suited representation. The interest of this algorithm is that it fits particularly well the architecture of modern computers allowing…
The Steiner tree problem aims to determine a minimum edge-weighted tree that spans a given set of terminal vertices from a given graph. In the past decade, a considerable number of algorithms have been developed to solve this…
This manuscript introduces Diophantine labeling, a new way of labeling of the vertices for finite simple undirected graphs with some divisibility condition on the edges. Maximal graphs admitting Diophantine labeling are investigated and…
Probability estimation is one of the fundamental tasks in statistics and machine learning. However, standard methods for probability estimation on discrete objects do not handle object structure in a satisfactory manner. In this paper, we…
An identifying code of a closed-twin-free graph $G$ is a dominating set $S$ of vertices of $G$ such that any two vertices in $G$ have a distinct intersection between their closed neighborhoods and $S$. It was conjectured that there exists…