Linear Forest Research Articles

Linear arboricity of degenerate graphs

AbstractA linear forest is a union of vertex‐disjoint paths, and the linear arboricity of a graph , denoted by , is the minimum number of linear forests into which the edge set of can be partitioned. Clearly, for a graph with maximum degree . On the other hand, the Linear Arboricity Conjecture due to Akiyama, Exoo, and Harary from 1981 asserts that for every graph . This conjecture has been verified for planar graphs and graphs whose maximum degree is at most , or is equal to or . Given a positive integer , a graph is ‐degenerate if it can be reduced to a trivial graph by successive removal of vertices with a degree at most . We prove that for any ‐degenerate graph provided .

The Role of High Nature Value Farmland for Landscape and Soil Pollution Assessment in a Coastal Delta in China Based on High-Resolution Indicators

High nature value farmland (HNVf) plays an important role in improving biodiversity and landscape heterogeneity, and it is effective in curbing soil non-point source pollution and carbon loss in sustainable eco-agricultural systems. To this end, we developed high-resolution (2 m × 2 m) indicators for the identification of potential HNVf based on GF1B remote sensing imaging, including the land cover (LC), normalized difference vegetation index (NDVI), Shannon diversity (SH), and Simpsons index (SI). The statistical results for LC with high resolution (2 m × 2 m) showed that there was 41.05% of intensive farmland in the study area, and the pixel proportion of the HNVf map (above G3) was 44.30%. These HNVf patches were concentrated in the transition zone around the edge of the intensive farmland and around rivers, with characteristics of HNVf type 2 being significantly reflected. Among the real-life areas from Map World, elements (i.e., linear forests, rivers, and semi-natural vegetation etc.) of HNVf accounted for more than 70% of these regions, while a field survey based on potential HNVf patches also exhibited significant HNVf characteristics in comparison with intensive farmlands. In addition, from 2002 to 2020, the total migration distance of the gravity center of intensive farmland in the study area was 7.65 km. Moreover, four landscape indices (patch COH index, landscape division index, SH, and SI) slowly increased, indicating that the species richness and biodiversity were improved. It was also found that a series of ecological protection policies provide effective guarantees for an improvement in species diversity and the development of HNVf in the study area. In particular, the average contents of As, Cr, Cu, Ni, and Zn in the HNVf were 20.99 mg kg−1, 121.11 mg kg−1, 21.97 mg kg−1, 29.34 mg kg−1, and 41.68 mg kg−1, respectively, which were lower in comparison with the intensive farmland soil. This is the first HNVf exploration for landscape and soil pollution assessment in a coastal delta in China, and could provide powerful guidance for the ecological protection of farmland soil and the high-quality development of sustainable agriculture.

Acyclic chromatic index of chordless graphs

An acyclic edge coloring of a graph is a proper edge coloring with no bichromatic cycles. The acyclic chromatic index of a graph G denoted by a′(G), is the minimum integer k such that G has an acyclic edge coloring with k colors. It was conjectured by Fiamčík [13] that a′(G)≤Δ+2 for any graph G with maximum degree Δ. Linear arboricity of a graph G, denoted by la(G), is the minimum number of linear forests into which the edges of G can be partitioned. A graph is said to be chordless if no cycle in the graph contains a chord. By a result of Basavaraju and Chandran [6], if G is chordless, then a′(G)≤Δ+1. Machado, de Figueiredo and Trotignon [23] proved that the chromatic index of a chordless graph is Δ when Δ≥3. We prove that for any chordless graph G, a′(G)=Δ, when Δ≥3. Notice that this is an improvement over the result of Machado et al., since any acyclic edge coloring is also a proper edge coloring and we are using the same number of colors. As a byproduct, we prove that la(G)=⌈Δ2⌉, when Δ≥3. To obtain the result on acyclic chromatic index, we prove a structural result on chordless graphs which is a refinement of the structure given by Machado et al. [23] in case of chromatic index. This might be of independent interest.

On the [formula omitted]-spectral radius of graphs without linear forests

Let A(G) and D(G) be the adjacency and degree matrices of a simple graph G on n vertices, respectively. The Aα-spectral radius of G is the largest eigenvalue of Aα(G)=αD(G)+(1−α)A(G) for a real number α∈[0,1]. In this paper, for α∈(0,1), we obtain a sharp upper bound for the Aα-spectral radius of graphs on n vertices without a subgraph isomorphic to a linear forest for n large enough and characterize all graphs which attain the upper bound. As a result, we completely dertermine the maximum signless Laplacian spectral radius of graphs on n vertices without a subgraph isomorphic to a linear forest for n large enough.

Non-Empty Intersection of Longest Paths in $H$-Free Graphs

We make progress toward a characterization of the graphs $H$ such that every connected $H$-free graph has a longest path transversal of size $1$. In particular, we show that the graphs $H$ on at most $4$ vertices satisfying this property are exactly the linear forests. We also show that if the order of a connected graph $G$ is large relative to its connectivity $\kappa(G)$, and its independence number $\alpha(G)$ satisfies $\alpha(G) \le \kappa(G) + 2$, then each vertex of maximum degree forms a longest path transversal of size $1$.

Preconditioning Sparse Matrices with Alternating and Multiplicative Operator Splittings

We present an algebraic framework for operator splitting preconditioners for general sparse matrices. The framework leads to four different approaches: two with alternating splittings and two with a multiplicative ansatz. The ansatz generalizes ADI and ILU methods to multiple factors and to a more general factor form. The factors may be computed directly from the matrix coefficients or adaptively by incomplete sparse inversions. The special case of tridiagonal splittings is examined in more detail. We decompose the adjacency graph of the sparse matrix into multiple (almost) disjoint linear forests, and each linear forest (a union of disjoint paths) leads to a tridiagonal splitting. We obtain specialized variants of the four general approaches. Parallel implementations for all steps are provided on a GPU. We demonstrate the effectiveness and efficiency of these preconditioners combined with GMRES on various matrices.

Partitioning Planar Graphs without 4-Cycles and 6-Cycles into a Linear Forest and a Forest

Partitioning Planar Graphs without 4-Cycles and 6-Cycles into a Linear Forest and a Forest

度和与边数条件下过线性森林圈的一些研究

度和与边数条件下过线性森林圈的一些研究

IMPORTANCE MEASUREMENT OF THE INFLUENCING FACTORS OF LONG-TERM NURSING STATUS IN LONG-TERM NURSING INSURANCE BASED ON MULTIPLE LINEAR REGRESSION, RANDOM FOREST AND XGBOOST MODELS

Long-term care for the elderly has become one of the prominent social problems globally when the ratios of persons whose ages over 65 steadily increase in almost all countries. One of the solution approaches that could be adapted is called long-term care insurance provided by insurance companies. However, companies need to classify care status types based on price or to provide supports utilizing its organizational structures such as departmental communication, business selection, and market segmentation since long-term care consists of many factors. The motivation of this research aims at filling the gap since there exists no comprehensive research concerning these factors that have impacts on the long-term care status for the elderly. To determine those factors, machine learning (ML) algorithms such as multiple linear regression, random forest, and the XGBoost are selected to be employed. Then, those factors and their important variables are utilized to predict insurance pricing. The 2018 Chinese (CHARLS) data set is used to determine factors that have key impacts on long-term care status in the elderly. Finally, all models are combined as a comprehensive model to generate better prediction accuracies innovatively. The results show that the three ML models can provide relatively consistent important measures of risk factors in determining the nursing status of the elderly. On the other hand, the prediction accuracy of the random forest and the XGBoost was improved by 0.6% and 1%, respectively, when compared to multiple linear regression. Besides, the results show that when the ratios of 2.6, 3.7, 3.7 are assigned to the results of the three models, the prediction accuracy of the comprehensive model is higher in the test set than that of the multiple linear regression, which contributes 1.92% more. The main innovation of this research is to construct a comprehensive model, a weighted combination of three models, with better prediction accuracy. Eventually, the long-term care insurance business can utilize the comprehensive model to classify the long-term care status of the elderly.

The Maximum Spectral Radius of Graphs without Spanning Linear Forests

The Maximum Spectral Radius of Graphs without Spanning Linear Forests

The Complexity of Routing Problems in Forbidden-Transition Graphs and Edge-Colored Graphs

The notion of forbidden-transition graphs allows for a robust generalization of walks in graphs. In a forbidden-transition graph, every pair of edges incident to a common vertex is permitted or forbidden; a walk is compatible if all pairs of consecutive edges on the walk are permitted. Forbidden-transition graphs and related models have found applications in a variety of fields, such as routing in optical telecommunication networks, road networks, and bio-informatics. A widely-studied special case are edge-colored graphs, where a compatible walk is forbidden to take two edges of the same color in a row. We initiate the study of fundamental problems on finding paths, cycles and walks in forbidden-transition graphs from the point of view of parameterized complexity, including an in-depth study of tractability with regards to various graph-width parameters. Among several results, we prove that finding a simple compatible path between given endpoints in a forbidden-transition graph is W[1]-hard when parameterized by the vertex-deletion distance to a linear forest (so it is also hard when parameterized by pathwidth or treewidth). On the other hand, we show an algebraic trick that yields tractability when parameterized by treewidth for finding a compatible Hamiltonian cycle in the edge-colored graph setting.

The linear arboricity of K5-minor free graphs

The linear arboricity l a ( G ) of a graph G is the minimum number of linear forests which partition the edges of G . In 1980, Akiyama et al. conjectured that for any graph G , ⌈ Δ ( G ) 2 ⌉ ≤ l a ( G ) ≤ ⌈ Δ ( G ) + 1 2 ⌉ . In the paper, we establish two essential structural properties of K 5 -minor free graphs G , one is used to confirm the conjecture for all K 5 -minor free graphs, the other is devoted to proving that l a ( G ) = ⌈ Δ ( G ) 2 ⌉ holds if Δ ( G ) ≥ 9 . In addition, we prove here that if G is a K 5 − -minor free graph and Δ ( G ) ≥ 5 , then l a ( G ) = ⌈ Δ ( G ) 2 ⌉ .

Further Results on the Generalized Turán Number of Spanning Linear Forests

Further Results on the Generalized Turán Number of Spanning Linear Forests

Maxima of the Q-index: Forbidden a fan

The complete split graph, denoted by Sn,k, is the graph obtained by joining a copy of Kk to n−k isolated vertices. As a special threshold graph, it plays an important role in graph theory and spectral graph theory. Some known results tell us that if q(G)>q(Sn,k) then G can contain many kinds of subgraphs, such as paths, linear forests, friendship graphs, flowers, and so on (where q(G) is the Q-index of G). Motivated by these results, we are interested in finding larger subgraphs in G. In this paper, we show that if q(G)>q(Sn,k) then G contains a fan H2k, where H2k denotes the graph obtained by joining a vertex to a path of order 2k. This implies several previous results, and a conjecture proposed by Li and Peng.

ASSESSMENT OF THE CURRENT STATE OF PROTECTIVE FOREST STRIPS OF VARIOUS WINDAGE

Forest strip systems have a multifunctional effect on agricultural landscapes, stabilize the agroecological situation, and form new sustainable agroforestry systems.
 Goal. On the basis of a comparative analysis of the taxation and reclamation parameters of the protective forest strips (PLP) of various windage to assess their current state.
 The novelty of the research lies in the assessment of the state of various types of ZLN consisting of curtains, wings, modular and linear forest strips on complex chestnut soils.
 Methods. The methodology included a survey of protective forest stands, laying of permanent test areas, determination of morphometric parameters taking into account the assortment of tree species and their condition. The width of the crown space, openwork, crown closeness coefficient, optical density, etc. were determined by the method of photographic images.
 Results. Most of the plantings were planted in the period from 1988-92 of the last century. Forest strips whose age exceeds 34 years were created in an ordinary way. In addition to the main breed, shrubs were present in the composition of the bands. Many years of experience in creating forest strips in the dry steppe has revealed the dependence of plantings on the conditions of the soil cover on which they are grown. Moreover, the successful survival of PZLP on meadow-chestnut soils is linked to depressions. Edaphic conditions become an important criterion when growing LP.
 Conclusion. When the edaphic conditions are significantly complicated, due to the complexity of soils and the rigidity of the climate, it is either extremely difficult or impossible to grow durable, stable and aerodynamically efficient tree plantations from tree species. In such cases, non-linear curtain and backstage plantings may be more reliable and effective.

Fault-free Hamiltonian paths passing through prescribed linear forests in balanced hypercubes with faulty links

The n-dimensional balanced hypercube BHn is one of candidate interconnection networks for multiprocessor systems, and it is a bipartite graph. Let F be a set of faulty links and L be a fault-free linear forest in BHn. For any two nodes u and v in different parts of BHn such that none of the paths in L has u or v as internal node or both of them as end nodes, we prove that BHn admits a fault-free hamiltonian path between u and v passing through L even if the total number of links in F and L is up to 2n−2. The upper bound on the number of links in F and L is optimal. This generalizes some known results.

The Maximum Number of Stars in a Graph Without Linear Forest

For two graphs J and H, the generalized Turán number, denoted by ex(n, J, H), is the maximum number of copies of J in an H-free graph of order n. A linear forest F is the disjoint union of paths. In this paper, we determine $$ex(n,S_r,F)$$ when n is large enough, which generalizes the results on $$ex(n, S_r, P_k)$$ and $$ex(n, (k+1) P_2)$$ . Finally, we prose a problem related to the number of graph copies in an F-free graph under shifting operations.

Hamiltonian Paths of -cubes Avoiding Faulty Links and Passing Through Prescribed Linear Forests

The <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula> -ary <inline-formula><tex-math notation="LaTeX">$n$</tex-math></inline-formula> -cube <inline-formula><tex-math notation="LaTeX">$Q_n^k$</tex-math></inline-formula> is one of the most attractive interconnection networks for parallel and distributed systems. Let <inline-formula><tex-math notation="LaTeX">$F$</tex-math></inline-formula> be a set of faulty links in <inline-formula><tex-math notation="LaTeX">$Q_n^k$</tex-math></inline-formula> and let <inline-formula><tex-math notation="LaTeX">$L$</tex-math></inline-formula> be a linear forest in <inline-formula><tex-math notation="LaTeX">$Q_n^k-F$</tex-math></inline-formula> such that <inline-formula><tex-math notation="LaTeX">$|E(L)|+|F|\leq 2n-3$</tex-math></inline-formula> . For any two distinct nodes <inline-formula><tex-math notation="LaTeX">$u$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$v$</tex-math></inline-formula> of <inline-formula><tex-math notation="LaTeX">$Q_n^k$</tex-math></inline-formula> with <inline-formula><tex-math notation="LaTeX">$n\geq 2$</tex-math></inline-formula> and odd <inline-formula><tex-math notation="LaTeX">$k\geq 3$</tex-math></inline-formula> , we prove that <inline-formula><tex-math notation="LaTeX">$Q_n^k-F$</tex-math></inline-formula> admits a Hamiltonian path between <inline-formula><tex-math notation="LaTeX">$u$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$v$</tex-math></inline-formula> passing through <inline-formula><tex-math notation="LaTeX">$L$</tex-math></inline-formula> if and only if none of the paths in <inline-formula><tex-math notation="LaTeX">$L$</tex-math></inline-formula> has <inline-formula><tex-math notation="LaTeX">$u$</tex-math></inline-formula> or <inline-formula><tex-math notation="LaTeX">$v$</tex-math></inline-formula> as internal nodes or both of them as end-nodes. The upper bound <inline-formula><tex-math notation="LaTeX">$2n-3$</tex-math></inline-formula> on <inline-formula><tex-math notation="LaTeX">$|E(L)|+|F|$</tex-math></inline-formula> is optimal in the worst case. The main results in this paper generalized some known results.

Dynamic ensemble prediction of cognitive performance in spaceflight

During spaceflight, astronauts face a unique set of stressors, including microgravity, isolation, and confinement, as well as environmental and operational hazards. These factors can negatively impact sleep, alertness, and neurobehavioral performance, all of which are critical to mission success. In this paper, we predict neurobehavioral performance over the course of a 6-month mission aboard the International Space Station (ISS), using ISS environmental data as well as self-reported and cognitive data collected longitudinally from 24 astronauts. Neurobehavioral performance was repeatedly assessed via a 3-min Psychomotor Vigilance Test (PVT-B) that is highly sensitive to the effects of sleep deprivation. To relate PVT-B performance to time-varying and discordantly-measured environmental, operational, and psychological covariates, we propose an ensemble prediction model comprising of linear mixed effects, random forest, and functional concurrent models. An extensive cross-validation procedure reveals that this ensemble outperforms any one of its components alone. We also identify the most important predictors of PVT-B performance, which include an individual's previous PVT-B performance, reported fatigue and stress, and temperature and radiation dose. This method is broadly applicable to settings where the main goal is accurate, individualized prediction of human behavior involving a mixture of person-level traits and irregularly measured time series.

HEALTH INSURANCE PREMIUM PREDICTION USING BLOCKCHAIN TECHNOLOGY AND RANDOM FOREST REGRESSION ALGORITHM

Blockchain technology is based on a sequence of blocks, where each block carries a certain amount of information. Medical records can be cryptographically secured in the health insurance ecosystem with blockchain technology. Here, blockchain technology model is used to create a user interface for storing data block wise. Also, Insurance premium is predicted using Support Vector Regression, Lasso Regression, Ridge Regression, Multiple Linear Regression and Random Forest Regression algorithms. Out of all these algorithms, Multiple Linear Regression algorithm gives the better result.