Vertices Of Tree Research Articles

Tree Covers of Graphs and their Linear Preservers

Let G be an undirected graph. A \textit{vertex tree cover} of G is a collection of trees such that every vertex of G is incident with at least one tree. Similarly, an edge tree cover is a collection of trees such that every edge of G is contained in at least one tree. The tree cover number is defined as the minimum number of trees required in such a cover. In this paper, we demonstrate that when considering specific types of tree covers, only vertex permutations act as linear operators that preserve the tree cover number of G .

G2-SCANN: Gaussian-kernel graph-based SLD clustering algorithm with natural neighbourhood

For most clustering methods, not only the number of clusters must be set in advance, but also various hyperparameters such as initial centroids, number of nearest neighbours, the minimum number of points, neighbourhood radius, and cutoff distance all require pre-specification. As one of the most promising unsupervised learning methods in machine intelligence, existing clustering methods cannot simultaneously handle datasets with arbitrary shapes, different densities, distinct sizes, and overlapping. Background outliers and high dimensionality make clustering problems more challenging. In this paper, we propose a novel universal clustering methodology, called G2-SCANN, which yields the best clustering performance for all 30 synthetic and real datasets without any hyperparameter tuning if the exact number of clusters is known. Firstly, the shortest path length (SPL) in complex network or graph-based geodesic distance is used to give a locally backbone-structured description of graph vertex similarity. Accordingly, SPL-weighted local degree (SLD) is defined as vertex attributes of a SPL-weighted graph expressed by G2-SPL adjacency matrix with ε-natural neighbourhood. Secondly, the process of calculating SLD for every data point in a bottom-up way directly leads to division from a complete graph constituted by all data points to a group of SLD trees. This brings the interpretability and the elimination of lone trees. Thirdly, contrastive learning of largest SLD values for finding root vertices of each divisive tree is conducted and top-down category message is then transmitted from the root vertices to all the leaf ones of a SLD tree. It eventually produces tree-like clusters. Totally, the proposed G2-SCANN method leverages both local neighbouring similarity of data points and global information about data distribution and makes it perform better than other methods.

Theoretical Framework for Virtual Logistics Centers Creation

Intermodal terminals and warehouses operate in different countries and deliver specific services to their customers. For many clients, it is important to receive a full set of the logistics services delivered by a single operator. However, individual intermodal terminals and warehouses may face challenges with providing these services, e.g., just-in-time goods delivery, goods distribution, cargo handling in non-standard situations, and others. In such cases, the cooperation between logistics companies may be required to organize the comprehensive service of cargo within supply chains. One of the possible solutions is to integrate transport and logistics services providers, establishing their cooperation within one virtual logistics center. The aim of this article is to justify theoretically the possibility of creating such a center by combining services performed by the intermodal terminals and warehouses already in operation under a single entity, in order to minimize the cost of logistics services and the time of goods delivery, as well as to create a comprehensive range of logistics services needed by customers. The relevance of the article and the novelty of the idea are associated with justification of the possibility of combining the activities of intermodal terminals and warehouses located separately in the region in order to improve the logistical service of customers. The theoretical basis for creating a virtual logistics center is based on graph theory methods. The article presents a theoretical model, based on a system of edges and vertices of the graph tree, which corresponds to the activities performed by separately located intermodal terminals and individual warehouses. The discussion is focused on the current problems of creating virtual logistics centers. The research results may be interesting for the managers of intermodal terminals, warehouses, and logistics centers, as well as other decision-makers involved in supply chains implementation and development.

Поиск множества допустимых управленческих решений в технических системах

Scientific and technological progress leads to a continuous increase in the number and complexity of designed systems, as a result of which there is a need to create modern information and intelligent decision support systems. Such systems will make it possible to predict various scenarios of management decisions, which in turn helps to increase the efficiency and quality of the project. Effective use of automation in the search for management decisions can reduce the time spent on analysis, improve the quality of decisions made and increase the overall productivity and competitiveness of the organization. The purpose of this work is to substantiate the scientific approach to the issue of automated search for acceptable management solutions in technical systems. To describe management decisions, a concept is proposed that allows one to compactly depict the structure of complex hierarchical systems. Many solutions are described, and the problem of finding acceptable management solutions is formulated. A system for automated search for management decisions is described. An algorithm is presented for searching solutions that are feasible with respect to a certain condition from a set of possible ones, based on removing the vertices of the search tree that are invalid according to the specification. A description of the automation of building a model of many technical solutions is presented. Using a model to search for a set of acceptable management decisions allows you to more accurately predict various scenarios for the management decision made and helps to increase the efficiency and improve the quality of the project.

Estimating the Aboveground Biomass of Robinia pseudoacacia Based on UAV LiDAR Data

Robinia pseudoacacia is widely planted in the Loess Plateau as a major soil and water conservation tree species because of its dense canopy, complex structure, and strong soil and water conservation ability. The precise measurement of small-scale locust forest biomass is crucial to monitoring and evaluating the carbon sequestration functions of soil and water conservation vegetation. This study focuses on an artificial locust forest planted in the early 1990s in Caijiachuan Basin, Jixian County, Shanxi Province. A drone equipped with LiDAR was used to obtain point cloud data and generate a canopy height model. A watershed segmentation algorithm was used to identify tree vertices and extract individual trees. A relationship model between tree height, diameter at breast height, and biomass, combined with sample survey data, was established to explore the spatial distribution of biomass in the artificial locust forest at the level of the entire basin. The results show the following: (1) the structural parameters of locust extracted using UAV point cloud data have a good degree of fit and accuracy, and the recall rate is 72.7%; (2) the average error rate of the extracted maximum tree height value of locust is 7%, that of the minimum tree height value is 14%, and that of the average tree height value is 18%; (3) the average error rate of the extracted maximum diameter at breast height of locust is 15%, that of the minimum diameter at breast height is 37%, and that of the average diameter at breast height is 36%; and (4) the average error rate of the biomass estimation of locust calculated using point cloud data is 16.0%.

Retracted: On Vertex Degree-Based Topological Indices for Fixed Branching Vertices of Trees

Retracted: On Vertex Degree-Based Topological Indices for Fixed Branching Vertices of Trees

Relative timing information and orthology in evolutionary scenarios

BackgroundEvolutionary scenarios describing the evolution of a family of genes within a collection of species comprise the mapping of the vertices of a gene tree T to vertices and edges of a species tree S. The relative timing of the last common ancestors of two extant genes (leaves of T) and the last common ancestors of the two species (leaves of S) in which they reside is indicative of horizontal gene transfers (HGT) and ancient duplications. Orthologous gene pairs, on the other hand, require that their last common ancestors coincides with a corresponding speciation event. The relative timing information of gene and species divergences is captured by three colored graphs that have the extant genes as vertices and the species in which the genes are found as vertex colors: the equal-divergence-time (EDT) graph, the later-divergence-time (LDT) graph and the prior-divergence-time (PDT) graph, which together form an edge partition of the complete graph.ResultsHere we give a complete characterization in terms of informative and forbidden triples that can be read off the three graphs and provide a polynomial time algorithm for constructing an evolutionary scenario that explains the graphs, provided such a scenario exists. While both LDT and PDT graphs are cographs, this is not true for the EDT graph in general. We show that every EDT graph is perfect. While the information about LDT and PDT graphs is necessary to recognize EDT graphs in polynomial-time for general scenarios, this extra information can be dropped in the HGT-free case. However, recognition of EDT graphs without knowledge of putative LDT and PDT graphs is NP-complete for general scenarios. In contrast, PDT graphs can be recognized in polynomial-time. We finally connect the EDT graph to the alternative definitions of orthology that have been proposed for scenarios with horizontal gene transfer. With one exception, the corresponding graphs are shown to be colored cographs.

An infinite-dimensional nonlinear equation related to Gibbs measures of a SOS model

For the solid-on-solid (SOS) model with an external field and with spin values from the set of all integers on a Cayley tree, each (gradient) Gibbs measure corresponds to a boundary law (an infinite-dimensional vector function defined on vertices of the Cayley tree) satisfying a nonlinear functional equation. Recently some translation-invariant and height-periodic (non-normalizable) solutions to the equation are found. Here, our aim is to find non-height-periodic and non-normalizable boundary laws for the SOS model. By such a solution one can construct a non-probability Gibbs measure. We find explicitly several non-normalizable boundary laws. Moreover, we reduce the problem to solving of a nonlinear, second-order difference equation. We give analytic and numerical analyses of the difference equation.

Kittel’s molecular zipper model on Cayley trees

Kittel’s 1D model represents a natural DNA with two strands as a (molecular) zipper, which may be separated as the temperature is varied. We define multidimensional version of this model on a Cayley tree and study the set of Gibbs measures. We reduce description of Gibbs measures to solving of a nonlinear functional equation, with unknown functions (called boundary laws) defined on vertices of the Cayley tree. Each boundary law defines a Gibbs measure. We give a general formula of free energy depending on the boundary law. Moreover, we find some concrete boundary laws and corresponding Gibbs measures. Explicit critical temperature for occurrence of a phase transition (non-uniqueness of Gibbs measures) is obtained.

Acyclic graphs with at least 2ℓ + 1 vertices are ℓ‐recognizable

AbstractThe ‐deck of an ‐vertex graph is the multiset of subgraphs obtained from it by deleting vertices. A family of ‐vertex graphs is ‐recognizable if every graph having the same ‐deck as a graph in the family is also in the family. We prove that the family of ‐vertex graphs with no cycles is ‐recognizable when (except for ). As a consequence, the family of ‐vertex trees is ‐recognizable when and . It is known that this fails when .

Urban Treetop Detection and Tree-Height Estimation from Unmanned-Aerial-Vehicle Images

Individual tree detection for urban forests in subtropical environments remains a great challenge due to the various types of forest structures, high canopy closures, and the mixture of evergreen and deciduous broadleaved trees. Existing treetop detection methods based on the canopy-height model (CHM) from UAV images cannot resolve commission errors in heterogeneous urban forests with multiple trunks or strong lateral branches. In this study, we improved the traditional local-maximum (LM) algorithm using a dual Gaussian filter, variable window size, and local normalized correlation coefficient (NCC). Specifically, we adapted a crown model of maximum/minimum tree-crown radii and an angle strategy to detect treetops. We then removed and merged the pending tree vertices. Our results showed that our improved LM algorithm had an average user accuracy (UA) of 87.3% (SD± 4.6), an average producer accuracy (PA) of 82.8% (SD± 4.1), and an overall accuracy of 93.3% (SD± 3.9) for sample plots with canopy closures less than 0.5. As for the sample plots with canopy closures from 0.5 to 1, the accuracies were 78.6% (SD± 31.5), 73.8% (SD± 10.3), and 68.1% (SD± 12.7), respectively. The tree-height estimation accuracy reached more than 0.96, with an average RMSE of 0.61 m. Our results show that the UAV-image-derived CHM can be used to accurately detect individual trees in mixed forests in subtropical cities like Shanghai, China, to provide vital tree-structure parameters for precise and sustainable forest management.

P2M: A Fast Solver for Querying Distance from Point to Mesh Surface

Most of the existing point-to-mesh distance query solvers, such as Proximity Query Package (PQP), Embree and Fast Closest Point Query (FCPW), are based on bounding volume hierarchy (BVH). The hierarchical organizational structure enables one to eliminate the vast majority of triangles that do not help find the closest point. In this paper, we develop a totally different algorithmic paradigm, named P2M , to speed up point-to-mesh distance queries. Our original intention is to precompute a KD tree (KDT) of mesh vertices to approximately encode the geometry of a mesh surface containing vertices, edges and faces. However, it is very likely that the closest primitive to the query point is an edge e (resp., a face f ), but the KDT reports a mesh vertex υ instead. We call υ an interceptor of e (resp., f ). The main contribution of this paper is to invent a simple yet effective interception inspection rule and an efficient flooding interception inspection algorithm for quickly finding out all the interception pairs. Once the KDT and the interception table are precomputed, the query stage proceeds by first searching the KDT and then looking up the interception table to retrieve the closest geometric primitive. Statistics show that our query algorithm runs many times faster than the state-of-the-art solvers.

Tolerant Radon partitions on the all-paths convexity in graphs

In a graph G, the all-paths transit function A(u, v), consists of the set of all vertices in the graph G which lies in some path connecting u and v. Convexity obtained by the all-paths transit function is called all-paths convexity. A partition of a set P of vertices of a graph G into two disjoint non-empty subsets is called a Radon partition if the convex hulls of the subsets intersect. A Radon partition (P0, Q0) of P into nonempty subsets is called t-tolerant Radon partition, if for any set S ⊆ P with | S | ≤ t , the intersection of the convex hulls 〈 P 0 − S 〉 ∩ 〈 Q 0 − S 〉 ≠ ∅ . Here we study the t-tolerant Radon number of G with respect to all-paths convexity. It is the minimum number k such that any collection of k vertices of G has t-tolerant Radon partition. We prove that if G has only one block, then k = 2 t + 2 whenever t ≥ 1 . Finally, we prove that if the block-cut vertex tree representation of G is a path then k = 2 t + 3 , and if the block-cut vertex tree representation of G is a tree that is not a path then k = 2 t + 4 .

An Individual Tree Segmentation Method That Combines LiDAR Data and Spectral Imagery

The dynamic monitoring of forest resources is an integral component of forest resource management and forest eco-system stability maintenance. In recent years, LiDAR (Light Detection and Ranging) has been increasingly utilized in precision forest surveys due to its great penetrating ability and capacity to detect forest vertical structure information. However, the present airborne LiDAR data individual tree segmentation algorithms are not highly adaptable to forest types, particularly in mixed coniferous and broad-leaved forest zones, where the accuracy of individual tree extraction is low, and trees are incorrectly recognized and missed. In order to address these issues, in this study, spectral images and LiDAR data of a red pine conifer–broadleaf mixed forest in the Changbai Mountain Nature Reserve in Jilin Province were chosen, and the normalized point cloud was segmented iteratively using the distance-threshold-based individual tree segmentation method to obtain the initial segmented individual tree vertices. For individual trees with deviations in the initial vertex identification position, and unidentified individual trees, identification anchor points of real tree vertices are added within the canopy of the trees. These identification anchor points have strong position directivity in LiDAR data, which can mark the individual trees whose vertices were misidentified or missed during the initial individual tree segmentation process and identify these two tuples. The tree vertices may be inserted precisely based on the 3D shape of the individual tree point cloud, and the seed-point-based individual tree segmentation method is used to segment the normalized point cloud and finish the extraction of individual trees in red pine mixed conifer forests. The results indicate that, compared to the previous individual tree segmentation approach based on the relative spacing between individual trees, this study enhances the accuracy of individual tree segmentation from 83% to 96%. The extremely high segmentation accuracy indicates that the proposed method can accurately identify individual trees based on remote sensing techniques to segment forest individual trees, can provide a data basis for subsequent individual tree information extraction, and has great potential in practical applications.

Two inverse eigenvalue problems for matrices whose graphs are trees

Inverse eigenvalue problem refers to the problem of reconstructing a matrix of a desired structure from a prescribed eigendata. In this paper, we discuss two additive inverse eigenvalue problems for matrices whose graph is tree. In order to analyze the problems, the vertices of the given tree are labeled in a suitable way so that concrete recurrence relations between the characteristic polynomials of the leading principal submatrices of the matrix can be obtained in a simple form. The method of obtaining the entries of the required matrix is constructive and provides an algorithm for computing the same. We provide some numerical examples to illustrate the results. The computations are done using SCILAB by feeding the eigendata and the adjacency pattern of the tree as inputs.

Morphing tree drawings in a small 3D grid

We study crossing-free grid morphs for planar drawings of trees using the third dimension. A morph consists of morphing steps, where vertices of the tree move simultaneously along straight-line trajectories at constant speeds. The aim is to find a morph between two given drawings of the same tree that has a small number of steps and is crossing-free, i.e. no two vertices overlap or no two edges of the tree intersect except in their common endpoints at any moment during the morph. It is already known, that there exists a crossing-free morph between two drawings of an $n$-vertex planar graph $G$ with $\mathcal{O}(n)$ morphing steps, and that using the third dimension the number of steps can be reduced to $\mathcal{O}(\log n)$ for an $n$-vertex tree. However, these morphs do not bound one practical parameter, the resolution. Can the number of steps still be reduced substantially by using the third dimension with an additional restriction of keeping the resolution bounded throughout the morph? We answer this question in affirmative by presenting a $3D$ non-crossing morph between two planar grid drawings of an $n$-vertex tree in $\mathcal{O}(\sqrt{n} \log n)$ morphing steps such that each intermediate drawing is a grid drawing, i.e., vertices of the tree are mapped to the nodes of a $3D$ grid of polynomial volume.

A new class of generative classifiers based on staged tree models

Generative models for classification use the joint probability distribution of the class variable and the features to construct a decision rule. Among generative models, Bayesian networks and naive Bayes classifiers are the most commonly used and provide a clear graphical representation of the relationship among all variables. However, these have the disadvantage of highly restricting the type of relationships that could exist, by not allowing for context-specific independence. Here we introduce a new class of generative classifiers, called staged tree classifiers, which formally account for context-specific independence. They are constructed by a partitioning of the vertices of an event tree from which conditional independence can be formally read. The naive staged tree classifier is also defined, which extends the classic naive Bayes classifier whilst retaining the same complexity. An extensive simulation study shows that the classification accuracy of staged tree classifiers is competitive with that of state-of-the-art classifiers and an example showcases their use in practice.

Some extremal problems on the distance involving peripheral vertices of trees with given matching number

Some extremal problems on the distance involving peripheral vertices of trees with given matching number

Parking on the infinite binary tree

Let $$(A_u : u \in \mathbb {B})$$ be i.i.d. non-negative integers that we interpret as car arrivals on the vertices of the full binary tree $$ \mathbb {B}$$ . Each car tries to park on its arrival node, but if it is already occupied, it drives towards the root and parks on the first available spot. It is known (Bahl et al. in Parking on supercritical Galton–Watson trees, arXiv:1912.13062 , 2019; Goldschmidt and Przykucki in Comb Probab Comput 28:23–45, 2019) that the parking process on $$ \mathbb {B}$$ exhibits a phase transition in the sense that either a finite number of cars do not manage to park in expectation (subcritical regime) or all vertices of the tree contain a car and infinitely many cars do not manage to park (supercritical regime). We characterize those regimes in terms of the law of A in an explicit way. We also study in detail the critical regime as well as the phase transition which turns out to be “discontinuous”.

Stationary solutions and local equations for interacting diffusions on regular trees

We study the invariant measures of infinite systems of stochastic differential equations (SDEs) indexed by the vertices of a regular tree. These invariant measures correspond to Gibbs measures associated with certain continuous specifications, and we focus specifically on those measures which are homogeneous Markov random fields. We characterize the joint law at any two adjacent vertices in terms of a new two-dimensional SDE system, called the “local equation”, which exhibits an unusual dependence on a conditional law. Exploiting an alternative characterization in terms of an eigenfunction-type fixed point problem, we derive existence and uniqueness results for invariant measures of the local equation and infinite SDE system. This machinery is put to use in two examples. First, we give a detailed analysis of the surprisingly subtle case of linear coefficients, which yields a new way to derive the famous Kesten-McKay law for the spectral measure of the regular tree. Second, we construct solutions of tree-indexed SDE systems with nearest-neighbor repulsion effects, similar to Dyson’s Brownian motion.