Edge Subdivision Research Articles

Maximizing vertex rescue: investigating the firefighter problem with edge subdivision

The firefighter problem is a game played on a connected graph where a fire breaks out at a vertex. In each round, a firefighter chooses a vertex to protect, and the fire then spreads to all unprotected neighbors of the burning vertex. This sequence continues until the fire can no longer spread. The objective is to maximize the number of saved vertices, necessitating a strategic approach by the firefighter. This problem serves as a simplistic model for scenarios such as disease spread in a network or people’s movement under specific circumstances. To simulate a scenario with limited resources, we introduce a new firefighter method, involving the protection of one vertex and subdividing one edge. This method, achieved by adding a new vertex to the edge, effectively delays the burning of the vertex connected to the edge, providing additional time for the firefighter to protect it. Our investigation delves into the firefighter problem with subdividing edges, focusing on specific graphs. Notably, we present an optimal strategy for a graph with a maximum degree at most three.

Volume Estimation of Stem Segments Based on a Tetrahedron Model Using Terrestrial Laser Scanning Data

Stem volume is a very important parameter in forestry inventory and carbon storage. The stem volume estimated by most existing methods deviates from its true value because the irregularity of the stem is usually overlooked. In this study, we propose a stem segment volume estimation based on the tetrahedron model using TLS data. First, the initial stem segment surface model, including the lower, upper, and outer triangular surface models, was gradually reconstructed. Next, the outer surface model was subdivided based on the edge subdivision. Then, a closed triangular surface model without self-intersection was obtained. Afterward, a tetrahedron model of the stem segment was generated using TetGen software (Version 1.6.0) for the triangular surface model. Finally, the stem segment volume was calculated by summing the volumes of all the tetrahedrons in the tetrahedron model. An experiment with 76 stem segments from different tree species with different parameters showed that the reconstructed stem segment surface model effectively reflected the geometrical features of the stem segment surface. Compared to the volume based on the simulated sectional measurement, the MAPE of the volume based on the tetrahedron model was 2.12%. The results demonstrated the validity of the presented method for stem surface reconstruction and stem volume estimation, and the volume based on the tetrahedron model was closer to the true value than that based on the sectional measurement.

Analysis of cable eccentric diameter measurement based on the production line

In order to meet the requirements of real-time monitoring and timely adjustment of cable diameter, a laser caliper is installed on the cable production line. Based on the laser diffraction measurement technology, the related laser diameter measuring instrument was developed. This equipment adopts a CCD array sensor and two mutually perpendicular pulse laser light sources, with a single exposure of only 0.1μs. It can identify the measuring objects with instantaneous changes, such as jittery workpieces, and obtain the outer diameter of the measured object through complex algorithm processing and relevant data calculation. The equipment is designed by relying on the laser diffraction measurement method and electromagnetic induction technology and using the CCD laser diffraction principle. In addition, the error can also be introduced into the linear array CCD image collector. TCD1703C is used for the linear array CCD, and the size length of each pixel is 7μm. If the measurement result is not processed by edge subdivision, the maximum measurement accuracy can only reach 7μm resolution.

Some results on the Aα-eigenvalues of a graph

Let G be a simple graph of order n. For α∈[0,1], the Aα-matrix of G is defined as Aα(G)=αD(G)+(1−α)A(G), where A(G) and D(G) are the adjacency matrix and the diagonal degree matrix of G, respectively. The eigenvalues of the Aα-matrix of G are called the Aα-eigenvalues of G. In this paper, we first study the properties on Aα-eigenvalues, i.e. how the Aα-eigenvalues behave under some kinds of graph transformations including vertex deletion, vertex contraction, edge deletion and edge subdivision. Moreover, we also present the relationships between the Aα-eigenvalues of G and its k-domination number, independence number, chromatic number and circumference, respectively.

Adaptive refinement for unstructured T-splines with linear complexity

We present an adaptive refinement algorithm for T-splines on unstructured 2D meshes. While for structured 2D meshes, one can refine elements alternatingly in horizontal and vertical direction, such an approach cannot be generalized directly to unstructured meshes, where no two unique global mesh directions can be assigned. To resolve this issue, we introduce the concept of direction indices, i.e., integers associated to each edge, which are inspired by theory on higher-dimensional structured T-splines. Together with refinement levels of edges, these indices essentially drive the refinement scheme. We combine these ideas with an edge subdivision routine that allows for I-nodes, yielding a very flexible refinement scheme that nicely distributes the T-nodes, preserving global linear independence, analysis-suitability (local linear independence) except in the vicinity of extraordinary nodes, sparsity of the system matrix, and shape regularity of the mesh elements. Further, we show that the refinement procedure has linear complexity in the sense of guaranteed upper bounds on a) the distance between marked and additionally refined elements, and on b) the ratio of the numbers of generated and marked mesh elements.

Research on Web Visual Development Platform Based on Microservice

In order to meet the needs of diversified terminal networks, a Web visual development platform based on microservices is proposed. According to the characteristics of network data distribution and resource attributes in the big data environment, select the service-oriented architecture as the basic data collaborative management system, establish a sample set containing various data types, calculate the density distribution of each type of data in the continuous function based on the sample set, and obtain the weight probability of the maximum and minimum density data of the data sample set by calculating the dispersion function. Finally, the weight of nodes in the visualization platform is established as the weight in the visualization matrix, and the high probability of edge subdivision is used to enhance the balance of edge development matrix. Comparative experiments show that the proposed method has strong applicability and high visualization effect for data with complex characteristics, data weight probability conforms to the expected value and Gaussian distribution, and the overall performance of the platform is excellent.

Total outer connected vertex-edge domination

A vertex [Formula: see text] of a graph is said to [Formula: see text]-[Formula: see text] dominate every edge incident with [Formula: see text], as well as every edge incident to vertices adjacent to [Formula: see text]. A subset [Formula: see text] is a total outer connected vertex-edge dominating set of a graph [Formula: see text] if every edge in [Formula: see text] is [Formula: see text]-[Formula: see text] dominated by a vertex in [Formula: see text], the subgraph induced by [Formula: see text] has no isolated vertices and the subgraph induced by [Formula: see text] is connected. We initiate the study of total outer connected vertex-edge domination in graphs. We show that the decision problem for total outer-connected vertex-edge domination problem is [Formula: see text]-Complete even for bipartite graphs. We prove that for every tree of order [Formula: see text] with [Formula: see text] leaves, [Formula: see text] and characterize the trees attaining the lower bound. We also study the effect of edge removal, edge addition and edge subdivision on total outer connected vertex-edge domination number of a graph.

Restrained geodetic domination of edge subdivision graph

For a connected graph [Formula: see text], a set [Formula: see text] subset of [Formula: see text] is said to be a geodetic set if all vertices in G should lie in some [Formula: see text] geodesic for some [Formula: see text]. The minimum cardinality of the geodetic set is the geodetic number. In this paper, the authors discussed the geodetic number, geodetic domination number, and the restrained geodetic domination of the edge subdivision graph.

Complexity of Some Generalized Operations on Networks

The number of spanning trees in a network determines the totality of acyclic and connected components present within. This number is termed as complexity of the network. In this article, we address the closed formulae of the complexity of networks’ operations such as duplication (split, shadow, and vortex networks of S n ), sum ( S n + W 3 , S n + K 2 , and C n ∘ K 2 + K 1 ), product ( S n ⊠ K 2 and W n ∘ K 2 ), semitotal networks ( Q S n and R S n ), and edge subdivision of the wheel. All our findings in this article have been obtained by applying the methods from linear algebra, matrix theory, and Chebyshev polynomials. Our results shall also be summarized with the help of individual plots and relative comparison at the end of this article.

König Graphs with Respect to the 4-Path and Its Spanning Supergraphs

We describe the class of graphs whose every subgraph has the next property: The maximal number of disjoint 4-paths is equal to the minimal cardinality of sets of vertices such that every 4-path in the subgraph contains at least one of these vertices.We completely describe the set of minimal forbidden subgraphs for this class. Moreover, we present an alternative description of the class based on the operations of edge subdivision applied to bipartite multigraphs and the addition of the so-called pendant subgraphs, isomorphic to triangles and stars.

Influence of the Edge Removal, Edge Addition and Edge Subdivision on the Double Vertex–Edge Domination Number of a Graph

A vertex v of a graph $$G=(V,E)$$ is said to ve-dominate every edge incident to v, as well as every edge adjacent to these incident edges. A set $$S \subseteq V$$ is a vertex–edge dominating set (double vertex–edge dominating set, respectively) if every edge of E is ve-dominated by at least one vertex (at least two vertices) of S. The minimum cardinality of a vertex–edge dominating set (double vertex–edge dominating set, respectively) of G is the vertex–edge domination number $$\gamma _{ve}(G)$$ (the double vertex–edge domination number $$\gamma _{dve}(G)$$ , respectively). The influence of edge removal, edge addition and edge subdivision on the double vertex–edge domination number of a graph are investigated in this paper.

Relations between edge removing and edge subdivision concerning domination number of a graph

Let e be an edge of a connected simple graph G. The graph obtained by removing (resp. subdividing) an edge e from G is denoted by G−e (resp. Ge). As usual, γ(G) denotes the domination number of G. We call G an SR-graph if for every edge e of G, γ(G−e)=γ(Ge); and G is an ASR-graph if for every edge e of G, γ(G−e)≠γ(Ge). In this work we give several examples of SR and ASR-graphs. We characterize SR-trees and show that ASR-graphs are efficient and γsd-critical. Consequently, ASR graphs are γ-insensitive and satisfy Vizing’s Conjecture.

Balanced subdivisions and flips on surfaces

In this paper, we show that two balanced triangulations of a closed surface are not necessarily connected by a sequence of balanced stellar subdivisions and welds. This answers a question posed by Izmestiev, Klee and Novik. We also show that two balanced triangulations of a closed surface are connected by a sequence of three local operations, which we call the pentagon contraction, the balanced edge subdivision and the balanced edge weld. In addition, we prove that two balanced triangulations of the 2 2 -sphere are connected by a sequence of pentagon contractions and their inverses if none of them are the octahedral sphere.

Topological features in crystal structures: a quotient graph assisted analysis of underlying nets and their embeddings

Topological properties of crystal structures may be analysed at different levels, depending on the representation and the topology that has been assigned to the crystal. Considered here is the combinatorial or bond topology of the structure, which is independent of its realization in space. Periodic nets representing one-dimensional complexes, or the associated graphs, characterize the skeleton of chemical bonds within the crystal. Since periodic nets can be faithfully represented by their labelled quotient graphs, it may be inferred that their topological features can be recovered by a direct analysis of the labelled quotient graph. Evidence is given for ring analysis and structure decomposition into building units and building networks. An algebraic treatment is developed for ring analysis and thoroughly applied to a description of coesite. Building units can be finite or infinite, corresponding to 1-, 2- or even 3-periodic subnets. The list of infinite units includes linear chains or sheets of corner- or edge-sharing polyhedra. Decomposing periodic nets into their building units relies on graph-theoretical methods classified as surgery techniques. The most relevant operations are edge subdivision, vertex identification, edge contraction and decoration. Instead, these operations can be performed on labelled quotient graphs, evidencing in almost a mechanical way the nature and connection mode of building units in the derived net. Various examples are discussed, ranging from finite building blocks to 3-periodic subnets. Among others, the structures of strontium oxychloride, spinel, lithiophilite and garnet are addressed.

Stellar Theory for Flag Complexes

Refining a basic result of Alexander, we show that two flag simplicial complexes are piecewise linearly homeomorphic if and only if they can be connected by a sequence of flag complexes, each obtained from the previous one by either an edge subdivision or its inverse. For flag spheres we pose new conjectures on their combinatorial structure forced by their face numbers, analogous to the extremal examples in the upper and lower bound theorems for simplicial spheres. Furthermore, we show that our algorithm to test the conjectures searches through the entire space of flag PL spheres of any given dimension.

On the growth of permutation classes

We study aspects of the enumeration of permutation classes, sets of permutations closed downwards under the subpermutation order. First, we consider monotone grid classes of permutations. We present procedures for calculating the generating function of any class whose matrix has dimensions m × 1 for some m, and of acyclic and unicyclic classes of gridded permutations. We show that almost all large permutations in a grid class have the same shape, and determine this limit shape. We prove that the growth rate of a grid class is given by the square of the spectral radius of an associated graph and deduce some facts relating to the set of grid class growth rates. In the process, we establish a new result concerning tours on graphs. We also prove a similar result relating the growth rate of a geometric grid class to the matching polynomial of a graph, and determine the effect of edge subdivision on the matching polynomial. We characterise the growth rates of geometric grid classes in terms of the spectral radii of trees. We then investigate the set of growth rates of permutation classes and establish a new upper bound on the value above which every real number is the growth rate of some permutation class. In the process, we prove new results concerning expansions of real numbers in non-integer bases in which the digits are drawn from sets of allowed values. Finally, we introduce a new enumeration technique, based on associating a graph with each permutation, and determine the generating functions for some previously unenumerated classes. We conclude by using this approach to provide an improved lower bound on the growth rate of the class of permutations avoiding the pattern 1324. In the process, we prove that, asymptotically, patterns in Łukasiewicz paths exhibit a concentrated Gaussian distribution.

Assessment of dynamic adaptive grids in Volume-Of-Fluid simulations of oblique drop impacts onto liquid films

Grid spacing dependence in three-dimensional numerical simulations of non-normal drop impact onto thin liquid films is assessed for different impingement angles and grid refinement levels. To describe the liquid phase dynamics, the Navier–Stokes equations are coupled to a Volume-Of-Fluid (VOF) model. Numerical simulations are performed with a modified version software OpenFOAM over a structured grid of hexaedra. Grid adaptation is carried out using an edge subdivision technique which results in non-conformal meshes. Grid convergence is assessed by monitoring integral parameters describing the dynamics of the post-impact free-surface waves. Starting from an initial grid spacing between D/8 and D/5, with D drop diameter, a refinement level of three is found to be sufficient to describe the diverse flow feature and to identify the splashing regime.

Growth Rates of Geometric Grid Classes of Permutations

Geometric grid classes of permutations have proven to be key in investigations of classical permutation pattern classes. By considering the representation of gridded permutations as words in a trace monoid, we prove that every geometric grid class has a growth rate which is given by the square of the largest root of the matching polynomial of a related graph. As a consequence, we characterise the set of growth rates of geometric grid classes in terms of the spectral radii of trees, explore the influence of “cycle parity” on the growth rate, compare the growth rates of geometric grid classes against those of the corresponding monotone grid classes, and present new results concerning the effect of edge subdivision on the largest root of the matching polynomial.

A Note on Aperiodic Ammann Tiles

We present a variant of Ammann tiles consisting of two similar rectilinear hexagons with edge subdivision, which can tile the plane but only in non-periodic ways. A special matching rule, ghost marking, plays a key role in the proof.

Zero forcing number, maximum nullity, and path cover number of subdivided graphs

The zero forcing number, maximum nullity and path cover number of a (simple, undirected) graph are parameters that are important in the study of minimum rank problems. We investigate the effects on these graph parameters when an edge is subdivided to obtain a so-called edge subdivision graph. An open question raised by Barrett et al. is answered in the negative, and we provide additional evidence for an affirmative answer to another open question in that paper [W. Barrett, R. Bowcutt, M. Cutler, S. Gibelyou, and K. Owens. Minimum rank of edge subdivisions of graphs. Electronic Journal of Linear Algebra, 18:530–563, 2009.]. It is shown that there is an independent relationship between the change in maximum nullity and zero forcing number caused by subdividing an edge once. Bounds on the effect of a single edge subdivision on the path cover number are presented, conditions under which the path cover number is preserved are given, and it is shown that the path cover number and the zero forcing number of a complete subdivision graph need not be equal.