Edge Intersection Research Articles

Intrinsic second-order topological insulators in two-dimensional polymorphic graphyne with sublattice approximation

In two dimensions, intrinsic second-order topological insulators (SOTIs) are characterized by topological corner states that emerge at the intersections of distinct edges with reversed mass signs, enforced by spatial symmetries. Here, we present a comprehensive investigation within the class BDI to clarify the symmetry conditions ensuring the presence of intrinsic SOTIs in two dimensions. We reveal that the (anti-)commutation relationship between spatial symmetries and chiral symmetry is a reliable indicator of intrinsic corner states. Through first-principles calculations, we identify several ideal candidates within carbon-based polymorphic graphyne structures for realizing intrinsic SOTIs under sublattice approximation. Furthermore, we show that the corner states in these materials persist even in the absence of sublattice approximation. Our findings not only deepen the understanding of higher-order topological phases but also open new pathways for realizing topological corner states that are readily observable.

Characterizations and Clique Coloring of Edge Intersection Graphs on a Triangular Grid

Characterizations and Clique Coloring of Edge Intersection Graphs on a Triangular Grid

On non-superperfection of edge intersection graphs of paths

The routing and spectrum assignment problem in modern flexgrid elastic optical networks asks for assigning to given demands a route in an optical network and a channel within an optical frequency spectrum so that the channels of two demands are disjoint whenever their routes share a link in the optical network. This problem can be modeled in two phases: firstly, a selection of paths in the network and, secondly, an interval coloring problem in the edge intersection graph of these paths. The interval chromatic number equals the smallest size of a spectrum such that a proper interval coloring is possible, the weighted clique number is a natural lower bound. Graphs where both parameters coincide for all possible non-negative integral weights are called superperfect. Therefore, the occurrence of non-superperfect edge intersection graphs of routing paths can provoke the need of larger spectral resources. In this work, we examine the question which minimal non-superperfect graphs can occur in the edge intersection graphs of routing paths in different underlying networks: when the network is a path, a tree, a cycle, or a sparse planar graph with small maximum degree. We show that for any possible network (even if it is restricted to a path) the resulting edge intersection graphs are not necessarily superperfect. We close with a discussion of possible consequences and of some lines of future research.

Experimental investigation on the impact of fuel injection schemes on the fuel/air mixing characteristics in alternating wedge struts

To enhance the performance of scramjet engines, it is essential to understand the fuel/air mixing characteristics downstream of alternating wedge struts and the dynamics of streamwise vortices. Two optical measurement techniques, Mie scattering and acetone-particle laser-induced fluorescence, were employed to study the impact of fuel injection location and different injection gases on the fuel/air mixing characteristics downstream of alternating wedge struts at total temperature of 300 K and total pressure of 800 kPa. The results indicated that the particle laser-induced fluorescence measurements provided superior results compared to Mie scattering, due to the more uniform distribution and better following characteristics of acetone molecules as seeding particles. Different fuel injection locations resulted in varying fuel distributions and plume morphologies downstream. The plume sizes obtained from the three types of fuel orifices were generally consistent, but the mixing effect of the orifices on the first strut was better, whose injection locations being at the trailing edge intersections, with a larger plume area and more uniform fuel distribution. The fuel/air mixing process downstream of the alternating wedge struts was primarily dominated by streamwise vortices. Despite certain differences in the physical properties of nitrogen and hydrogen molecules, the measurement results showed that the plume morphologies obtained with both gases were quite similar. Additionally, a streamwise vortex model was developed to predict the dynamics of streamwise vortices and the evolution of fuel plume morphology, and it was compared with experimental results.

Formulation, Proposition and Application of Interval Semigraph

Semigraph represents a versatile extension within graph theory, offering a broader framework for exploration. Among these, intersection graphs hold pivotal roles, serving as essential analytical tools with wide-ranging applications. In this study, we introduce the integration of intersection graphs into the generalized structure of semigraphs, aiming to invigorate researchers’ interest and encourage further advancements in this domain. This paper introduces two novel concepts: edge intersection semigraphs and interval semigraphs. Additionally, foundational results pertaining to these concepts are established, elucidating their fundamental properties. Furthermore, we illustrate the practical utility of interval semigraphs through an illustrative example involving their application in optimizing road traffic management systems.

Ecological network optimization of Jiuquan City, Gansu, China based on complex network theory and circuit model

Building a scientific and reasonable ecological network is the key for optimizing the pattern of territorial development and protection, and is of great significance for ensuring regional ecological security and promoting the virtuous cycle of ecosystems. In previous studies, nodal attack method (destruction of ecological source area) was often used in the "robustness" evaluation of ecological networks. Actually, the ecological corridor is more fragile than the source area, and thus the nodal attack method is not reasonable. In this study, taking Jiuquan City as the research area, based on the circuit model to construct the ecological network, we carried out the topology optimization of ecological network by using three strategies (random edge increase, node degree and priority edge increase with low node intermedium number) in complex network theory. We compared and analyzed the "robustness" of ecological network before and after optimization by constructing edge attack strategy, and selected the best network optimization strategy. The results showed that 65 ecological source areas were identified in Jiuquan City, with a total area of 20275.15 km2, and that grassland accounted for 89.5% of the source area. We identified 179 ecological corridors with a total length of 6387.16 km, 158 ecological barrier points with a total area of 1385.5 km2. The unused land accounted for 92.2% of the total barrier points area. We identified 63 ecological pinch points, mainly concentrated in the source edge and corridor intersection. Among them, the spatial distribution of 11 barrier points and pinch points was consistent, which was the key area to be repaired in ecological network optimization. The three optimization strategies had significantly improved the stability of ecological network in Jiuquan City. The relative size of the maximum connected subgraph and the edge connected rate of the ecological network of the optimization strategy of adding edges according to degree were all the most stable under random attack mode and deliberate attack mode, which was the best optimization scheme for ecological network in Jiuquan City.

Investigating the Efficiency of Using U-Net, Erf-Net and DeepLabV3 Architectures in Inverse Lithography-based 90-nm Photomask Generation

The paper deals with the inverse problem of computational lithography. We turn to deep neural network algorithms to compute photomask topologies. The chief goal of the research is to understand how efficient the neural net architectures such as U-net, Erf-Net and Deep Lab v.3, as well as built-in Calibre Workbench algorithms, can be in tackling inverse lithography problems. Specially generated and marked data sets are used to train the artificial neural nets. Calibre EDA software is used to generate haphazard patterns for a 90 nm transistor gate mask. The accuracy and speed parameters are used for the comparison. The edge placement error (EPE) and intersection over union (IOU) are used as metrics. The use of the neural nets allows two orders of magnitude reduction of the mask computation time, with accuracy keeping to 92% for the IOU metric.

On the j-Edge Intersection Graph of Cycle Graph

This paper defines a new class of graphs using the spanning subgraphs of a cycle graph as vertices. This class of graphs is called $j$-edge intersection graph of cycle graph, denoted by $E_{C_{(n,j)}}$. The vertex set of $E_{C_{(n,j)}}$ is the set of spanning subgraphs of cycle graph with $j$ edges where $n \geq 3$ and $j$ is a nonnegative integer such that $1 \leq j \leq n$. Moreover, two distinct vertices are adjacent if they have exactly one edge in common. $E_{C_{(n,j)}}$ is considered as a simple graph. Furthermore, $E_{C_{(n,j)}}$ is characterized by the value of $j$ that is when $j=1$ or $\lceil \frac{n}{2} \rceil < j \leq n$ and $2 \leq j \leq \lceil \frac{n}{2} \rceil$. When $j=1$ or $\lceil \frac{n}{2} \rceil < j \leq n$, the new graph only produced an empty graph. Hence, the proponents only considered the value when $2 \leq j \leq \lceil \frac{n}{2} \rceil$ in determining the order and size of $E_{C_{(n,j)}}$. Moreover, this paper discusses necessary and sufficient conditions where the $j$-edge intersection graph of $C_n$ is isomorphic to the cycle graph. Furthermore, the researchers determined a lower bound for the independence number, and an upper bound for the domination number of $E_{C_{(n,j)}}$ when $j=2$.

The crossing number of the generalized Petersen graph P(3k,k) in the projective plane

ABSTRACT The crossing number of a graph G in a surface Σ, denoted by c r Σ ( G ) , is the minimum number of pairwise intersections of edges in a drawing of G in Σ. Let k be an integer satisfying k ≥ 3 , the generalized Petersen graph P ( 3 k , k ) is the graph with vertex set V ( P ( 3 k , k ) ) = { u i , v i ∣ i = 1 , 2 , … , 3 k } and edge set E ( P ( 3 k , k ) ) = { u i u i + 1 , u i v i , v i v k + i ∣ i = 1 , 2 , … , 3 k } , the subscripts are read modulo 3 k . In this paper we investigate the crossing number of P ( 3 k , k ) in the projective plane. We determine the exact value of c r N 1 ( P ( 3 k , k ) ) is k–2 when 3 ≤ k ≤ 7 , moreover, for k ≥ 8 , we get that k − 2 ≤ c r N 1 ( P ( 3 k , k ) ) ≤ k − 1.

Hanani-Tutte for Radial Planarity II

A drawing of a graph $G$, possibly with multiple edges but without loops, is radial if all edges are drawn radially, that is, each edge intersects every circle centered at the origin at most once. $G$ is radial planar if it has a radial embedding, that is, a crossing-free radial drawing. If the vertices of $G$ are ordered or partitioned into ordered levels (as they are for leveled graphs), we require that the distances of the vertices from the origin respect the ordering or leveling.
 A pair of edges $e$ and $f$ in a graph is independent if $e$ and $f$ do not share a vertex. We show that if a leveled graph $G$ has a radial drawing in which every two independent edges cross an even number of times, then $G$ is radial planar. In other words, we establish the strong Hanani-Tutte theorem for radial planarity. This characterization yields a very simple algorithm for radial planarity testing.

PCP-Ed: Parallel coordinate plots for ensemble data

The Parallel Coordinate Plot (PCP) is a complex visual design commonly used for the analysis of high-dimensional data. Increasing data size and complexity may make it challenging to decipher and uncover trends and outliers in a confined space. A dense PCP image resulting from overlapping edges may cause patterns to be covered. We develop techniques aimed at exploring the relationship between data dimensions to uncover trends in dense PCPs. We introduce correlation glyphs in the PCP view to reveal the strength of the correlation between adjacent axis pairs as well as an interactive glyph lens to uncover links between data dimensions by investigating dense areas of edge intersections. We also present a subtraction operator to identify differences between two similar multivariate data sets and relationship-guided dimensionality reduction by collapsing axis pairs. We finally present a case study of our techniques applied to ensemble data and provide feedback from a domain expert in epidemiology.

On cycle graphs

The Cycle Graph [Formula: see text] of a graph [Formula: see text] is the edge intersection graph of all induced cycles of [Formula: see text]. In this paper, necessary and sufficient condition on [Formula: see text] such that [Formula: see text] is a connected graph, a tree, a bipartite graph is obtained. Non-existence of forbidden subgraph characterization for cycle graphs is also discussed. It is proved that the girth of cycle graph is three. The relationship between radius of [Formula: see text] and its [Formula: see text] is studied. Similar results on diameter and domination number of [Formula: see text] and its cycle graph are also obtained.

Monotonic Representations of Outerplanar Graphs as Edge Intersection Graphs of Paths on a Grid

In a representation of a graph $G$ as an edge intersection graph of paths on a grid (EPG) every vertex of $G$ is represented by a path on a grid and two paths share a grid edge iff the corresponding vertices are adjacent. In a monotonic EPG representation every path on the grid is ascending in both rows and columns. In a (monotonic) $B_k$-EPG representation every path on the grid has at most $k$ bends. The (monotonic) bend number $b(G)$ ($b^m(G)$) of a graph $G$ is the smallest natural number $k$ for which there exists a (monotonic) $B_k$-EPG representation of $G$. In this paper we deal with the monotonic bend number of outerplanar graphs and show that $b^m(G)\leqslant 2$ holds for every outerplanar graph $G$. Moreover, we characterize the maximal outerplanar graphs and the cacti with (monotonic) bend number equal to $0$, $1$ and $2$ in terms of forbidden induced subgraphs. As a byproduct we obtain low-degree polynomial time algorithms to construct (monotonic) EPG representations with the smallest possible number of bends for maximal outerplanar graphs and cacti.

Neural dual contouring

We introduce neural dual contouring (NDC), a new data-driven approach to mesh reconstruction based on dual contouring (DC). Like traditional DC, it produces exactly one vertex per grid cell and one quad for each grid edge intersection, a natural and efficient structure for reproducing sharp features. However, rather than computing vertex locations and edge crossings with hand-crafted functions that depend directly on difficult-to-obtain surface gradients, NDC uses a neural network to predict them. As a result, NDC can be trained to produce meshes from signed or unsigned distance fields, binary voxel grids, or point clouds (with or without normals); and it can produce open surfaces in cases where the input represents a sheet or partial surface. During experiments with five prominent datasets, we find that NDC, when trained on one of the datasets, generalizes well to the others. Furthermore, NDC provides better surface reconstruction accuracy, feature preservation, output complexity, triangle quality, and inference time in comparison to previous learned (e.g., neural marching cubes, convolutional occupancy networks) and traditional (e.g., Poisson) methods. Code and data are available at https://github.com/czq142857/NDC.

An Open-Source Processing Pipeline for Quad-Dominant Mesh Generation for Class-Compliant Ship Structural Analysis

We present an algorithm for the fully automatic generation of a class-compliant mesh for ship structural analysis. Our algorithm is implemented as an end-to-end solution. It starts from a description of a geometry and produces a class conforming surface mesh as a result. The algorithm consists of two parts, the automatic geometry refinement and the preconditioned Delaunay frontal quad dominant mesh generator. A geometry is described by a dictionary of elements and it contains points, rods, plates, and openings. A dictionary can contain modeling errors such as unintended overlaps or an unintended loss of connectivity between elements. The main contribution of the paper is the automatic geometry refinement algorithm and the virtual stiffener procedure designed to control the local mesh orientation of a marching front meshing algorithm. The geometry refinement algorithm guarantees that the output dictionary will be such that intersections of the boundary edges of plates are guaranteed to be nodes of any mesh generated by tessellating such geometry. The algorithm is implemented in Python, using the open-source Gmsh system together with the Open CASCADE kernel which is used to implement the automatic geometry refinement. We present several benchmark models from an engineering practice to illustrate our claims as well as to benchmark the efficiency of the various stages of the processing pipeline.

Data-Driven Modeling of Geometry-Adaptive Steady Heat Convection Based on Convolutional Neural Networks

Heat convection is one of the main mechanisms of heat transfer, and it involves both heat conduction and heat transportation by fluid flow; as a result, it usually requires numerical simulation for solving heat convection problems. Although the derivation of governing equations is not difficult, the solution process can be complicated and usually requires numerical discretization and iteration of differential equations. In this paper, based on neural networks, we developed a data-driven model for an extremely fast prediction of steady-state heat convection of a hot object with an arbitrary complex geometry in a two-dimensional space. According to the governing equations, the steady-state heat convection is dominated by convection and thermal diffusion terms; thus the distribution of the physical fields would exhibit stronger correlations between adjacent points. Therefore, the proposed neural network model uses convolutional neural network (CNN) layers as the encoder and deconvolutional neural network (DCNN) layers as the decoder. Compared with a fully connected (FC) network model, the CNN-based model is good for capturing and reconstructing the spatial relationships of low-rank feature spaces, such as edge intersections, parallelism, and symmetry. Furthermore, we applied the signed distance function (SDF) as the network input for representing the problem geometry, which contains more information compared with a binary image. For displaying the strong learning and generalization ability of the proposed network model, the training dataset only contains hot objects with simple geometries: triangles, quadrilaterals, pentagons, hexagons, and dodecagons, while the testing cases use arbitrary and complex geometries. According to the study, the trained network model can accurately predict the velocity and temperature field of the problems with complex geometries, which has never been seen by the network model during the model training; and the prediction speed is two orders faster than the CFD. The ability of accurate and extremely fast prediction of the network model suggests the potential of applying reduced-order network models to the applications of real-time control and fast optimization in the future.

Geodesic Hermite Spline Curve on Triangular Meshes

Curves on a polygonal mesh are quite useful for geometric modeling and processing such as mesh-cutting and segmentation. In this paper, an effective method for constructing C1 piecewise cubic curves on a triangular mesh M while interpolating the given mesh points is presented. The conventional Hermite interpolation method is extended such that the generated curve lies on M. For this, a geodesic vector is defined as a straightest geodesic with symmetric property on edge intersections and mesh vertices, and the related geodesic operations between points and vectors on M are defined. By combining cubic Hermite interpolation and newly devised geodesic operations, a geodesic Hermite spline curve is constructed on a triangular mesh. The method follows the basic steps of the conventional Hermite interpolation process, except that the operations between the points and vectors are replaced with the geodesic. The effectiveness of the method is demonstrated by designing several sophisticated curves on triangular meshes and applying them to various applications, such as mesh-cutting, segmentation, and simulation.

C-Planarity Testing of Embedded Clustered Graphs with Bounded Dual Carving-Width

For a clustered graph, i.e, a graph whose vertex set is recursively partitioned into clusters, the C-Planarity Testing problem asks whether it is possible to find a planar embedding of the graph and a representation of each cluster as a region homeomorphic to a closed disk such that (1) the subgraph induced by each cluster is drawn in the interior of the corresponding disk, (2) each edge intersects any disk at most once, and (3) the nesting between clusters is reflected by the representation, i.e., child clusters are properly contained in their parent cluster. The computational complexity of this problem, whose study has been central to the theory of graph visualization since its introduction in 1995 [Feng, Cohen, and Eades, Planarity for clustered graphs, ESA’95], has only been recently settled [Fulek and Tóth, Atomic Embeddability, Clustered Planarity, and Thickenability, to appear at SODA’20]. Before such a breakthrough, the complexity question was still unsolved even when the graph has a prescribed planar embedding, i.e, for embedded clustered graphs. We show that the C-Planarity Testing problem admits a single-exponential single-parameter FPT (resp., XP) algorithm for embedded flat (resp., non-flat) clustered graphs, when parameterized by the carving-width of the dual graph of the input. These are the first FPT and XP algorithms for this long-standing open problem with respect to a single notable graph-width parameter. Moreover, the polynomial dependency of our FPT algorithm is smaller than the one of the algorithm by Fulek and Tóth. In particular, our algorithm runs in quadratic time for flat instances of bounded treewidth and bounded face size. To further strengthen the relevance of this result, we show that an algorithm with running time O(r(n)) for flat instances whose underlying graph has pathwidth 1 would result in an algorithm with running time O(r(n)) for flat instances and with running time O(r(n^2) + n^2) for general, possibly non-flat, instances.

Coverage hole detection method of wireless sensor network based on clustering algorithm

The traditional method does not calculate the moving distance parameter when detecting coverage vulnerabilities, so its coverage performance is poor. The clustering algorithm can effectively calculate the moving distance parameter. Therefore, a method to detect the coverage hole in wireless sensor network based on clustering algorithm was proposed. Firstly, the parameter of coverage hole in wireless sensor network were calculated, including the moving distance parameters, circular intersection area parameters and redundancy parameters. Secondly, the algorithm of hole edge intersection was used to judge edge nodes of coverage holes in wireless sensor network. By judging whether there was a hole edge intersection on the sensing circle, the edge nodes of coverage holes in wireless sensor network could be determined. Thirdly, the mobile nodes were deployed in the way of airdrop, and then they were distributed evenly. The density of the initially deployed mobile nodes should meet the highest coverage requirement in wireless sensor network. After determining the edge nodes of coverage holes in wireless sensor network, the deployed mobile nodes were used to find the edge nodes of coverage holes in wireless sensor networks by random walk. Meanwhile, the coverage holes were approximated by the set of edge nodes. Finally, it was able to detect the coverage holes in wireless sensor network by clustering algorithm, including the detection of hole shape and the judgment of holes size. In order to verify the gap coverage performance of the coverage hole detection method based on clustering algorithm, the traditional method was compared with the proposed method. Experimental results show that the gap coverage performance of the proposed method is better than that of traditional method. The gap coverage performance of this method is 389 T, while the traditional method is only 351 T. so this method is more suitable for the detection of coverage hole in wireless sensor network.

Efficient Point-in-Polygon Tests by Grids Without the Trouble of Tuning the Grid Resolutions.

The grid-based approach is popular for point-in-polygon tests. However, there is a trade-off between the preprocessing and the inclusion test, which always requires the grid resolutions to be tuned. In this article, we address this challenge by enhancing the grid structure using y-axis-aligned stripes, which are formed by the y-axis-aligned lines passing through the endpoints of the edge segments in the cell, thereby managing the edge segments in each grid cell. Moreover, we precompute the inclusion properties of the x-axis-aligned top borders of the stripes during preprocessing. Therefore, to answer a query point with the ray crossing method, we can emit a ray from the point to propagate upwards until the ray arrives at the top border of a stripe. We thoroughly consider singular cases to guarantee each query point can be answered in the stripe that contains the point. In our method, the computational load can be decreased, as one coordinate of the intersection point between the ray and an edge is known in advance, and parallel computing can be well exploited because the branching operations for determining whether an edge intersects with the ray are saved. Experimental results show that the efficiency of our method does not vary much with respect to the grid resolutions, so the trouble of tuning grid resolutions can be avoided. Ultimately, our method with a low grid resolution can reduce the preprocessing time and still achieve a higher inclusion test efficiency than the existing methods with a high grid resolution, especially on GPUs.