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Journal articles on the topic 'Geometric Intersection Graphs'

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Author: Grafiati

Published: 6 September 2023

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1

Fekete, Sándor P., and Phillip Keldenich. "Conflict-Free Coloring of Intersection Graphs." International Journal of Computational Geometry & Applications 28, no. 03 (September 2018): 289–307. http://dx.doi.org/10.1142/s0218195918500085.

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A conflict-free[Formula: see text]-coloring of a graph [Formula: see text] assigns one of [Formula: see text] different colors to some of the vertices such that, for every vertex [Formula: see text], there is a color that is assigned to exactly one vertex among [Formula: see text] and [Formula: see text]’s neighbors. Such colorings have applications in wireless networking, robotics, and geometry, and are well studied in graph theory. Here we study the conflict-free coloring of geometric intersection graphs. We demonstrate that the intersection graph of [Formula: see text] geometric objects without fatness properties and size restrictions may have conflict-free chromatic number in [Formula: see text] and in [Formula: see text] for disks or squares of different sizes; it is known for general graphs that the worst case is in [Formula: see text]. For unit-disk intersection graphs, we prove that it is NP-complete to decide the existence of a conflict-free coloring with one color; we also show that six colors always suffice, using an algorithm that colors unit disk graphs of restricted height with two colors. We conjecture that four colors are sufficient, which we prove for unit squares instead of unit disks. For interval graphs, we establish a tight worst-case bound of two.

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2

Baste, Julien, and Dimitrios M. Thilikos. "Contraction Bidimensionality of Geometric Intersection Graphs." Algorithmica 84, no. 2 (January 24, 2022): 510–31. http://dx.doi.org/10.1007/s00453-021-00912-w.

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3

Eppstein, David. "Testing bipartiteness of geometric intersection graphs." ACM Transactions on Algorithms 5, no. 2 (March 2009): 1–35. http://dx.doi.org/10.1145/1497290.1497291.

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4

Cabello, Sergio, and Wolfgang Mulzer. "Minimum cuts in geometric intersection graphs." Computational Geometry 94 (March 2021): 101720. http://dx.doi.org/10.1016/j.comgeo.2020.101720.

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5

Uehara, Ryuhei. "Tractabilities and Intractabilities on Geometric Intersection Graphs." Algorithms 6, no. 1 (January 25, 2013): 60–83. http://dx.doi.org/10.3390/a6010060.

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6

Erlebach, Thomas, and Jiri Fiala. "On-line coloring of geometric intersection graphs." Computational Geometry 23, no. 2 (September 2002): 243–55. http://dx.doi.org/10.1016/s0925-7721(02)00089-5.

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7

Tokunaga, Shin-ichi. "Intersection number of two connected geometric graphs." Information Processing Letters 59, no. 6 (September 1996): 331–33. http://dx.doi.org/10.1016/0020-0190(96)00124-x.

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8

Erlebach, Thomas, Klaus Jansen, and Eike Seidel. "Polynomial-Time Approximation Schemes for Geometric Intersection Graphs." SIAM Journal on Computing 34, no. 6 (January 2005): 1302–23. http://dx.doi.org/10.1137/s0097539702402676.

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9

Agnarsson, Geir, Peter Damaschke, and Magnús M. Halldórsson. "Powers of geometric intersection graphs and dispersion algorithms." Discrete Applied Mathematics 132, no. 1-3 (October 2003): 3–16. http://dx.doi.org/10.1016/s0166-218x(03)00386-x.

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10

de Berg, Mark, Sándor Kisfaludi-Bak, and Gerhard Woeginger. "The complexity of Dominating Set in geometric intersection graphs." Theoretical Computer Science 769 (May 2019): 18–31. http://dx.doi.org/10.1016/j.tcs.2018.10.007.

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11

Kim, Seog-Jin, and Kittikorn Nakprasit. "Coloring the complements of intersection graphs of geometric figures." Discrete Mathematics 308, no. 20 (October 2008): 4589–94. http://dx.doi.org/10.1016/j.disc.2007.08.072.

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12

Cabello, Sergio, and Miha Jejčič. "Refining the Hierarchies of Classes of Geometric Intersection Graphs." Electronic Notes in Discrete Mathematics 54 (October 2016): 223–28. http://dx.doi.org/10.1016/j.endm.2016.09.039.

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13

Pawlik, Arkadiusz, Jakub Kozik, Tomasz Krawczyk, Michał Lasoń, Piotr Micek, William T. Trotter, and Bartosz Walczak. "Triangle-Free Geometric Intersection Graphs with Large Chromatic Number." Discrete & Computational Geometry 50, no. 3 (August 29, 2013): 714–26. http://dx.doi.org/10.1007/s00454-013-9534-9.

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14

Keller, Chaya, and Shakhar Smorodinsky. "Conflict-Free Coloring of Intersection Graphs of Geometric Objects." Discrete & Computational Geometry 64, no. 3 (June 3, 2019): 916–41. http://dx.doi.org/10.1007/s00454-019-00097-8.

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15

da Fonseca, Guilherme D., Vinícius Gusmão Pereira de Sá, and Celina Miraglia Herrera de Figueiredo. "Shifting Coresets: Obtaining Linear-Time Approximations for Unit Disk Graphs and Other Geometric Intersection Graphs." International Journal of Computational Geometry & Applications 27, no. 04 (December 2017): 255–76. http://dx.doi.org/10.1142/s0218195917500078.

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Numerous approximation algorithms for problems on unit disk graphs have been proposed in the literature, exhibiting a sharp trade-off between running times and approximation ratios. We introduce a variation of the known shifting strategy that allows us to obtain linear-time constant-factor approximation algorithms for such problems. To illustrate the applicability of the proposed variation, we obtain results for three well-known optimization problems. Among such results, the proposed method yields linear-time [Formula: see text]-approximations for the maximum-weight independent set and the minimum dominating set of unit disk graphs, thus bringing significant performance improvements when compared to previous algorithms that achieve the same approximation ratios. Finally, we use axis-aligned rectangles to illustrate that the same method may be used to derive linear-time approximations for problems on other geometric intersection graph classes.

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16

Kurniawan, Bayu, Helga Yermadona, and Surya Eka Priana. "EVALUASI SIMPANG TAK BERSINYAL DI SIMPANG LIMAU BUKITTINGGI." Ensiklopedia Research and Community Service Review 1, no. 3 (August 1, 2022): 20–28. http://dx.doi.org/10.33559/err.v1i3.1232.

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Simpang Limau is one of the Three Unsignalized Intersections on the Bukittinggi-Payakumbuh Cross Road, which has a fairly heavy traffic flow. The purpose of this study was to analyze the performance of an unsignaled intersection using MKJI 1997. The data needed in this study were primary data and secondary data. The primary data is in the form of data on traffic flow conditions, geometric intersections and environmental conditions. Secondary data includes population growth and road network maps. Traffic data collection is carried out by conducting a direct survey to the location for three days, namely Saturday, Sunday, and Monday from 07.00 to 18.00. This data was processed using Microsoft Excel based on the 1997 MKJI and the results obtained were made in the form of tables and graphs. From these tables and graphs it can be seen that the traffic volume, traffic composition, and peak hours that occur at the intersection. The results of calculations at peak hours with the highest flow, namely on Sundays at 17.00-18.00, the traffic volume at this intersection is 1,484 (pcu/hour). The traffic delay at the intersection is 3,066 (sec/pcu). The value of the degree of saturation in the existing condition of 0.74 does not exceed the MKJI determination of 0.75. The value of the intersection delay is 2.693 (sec/pcu) and the probability of the intersection queue is 22.32% to 71.68%.Key words : Unsignalized Intersection, Capacity (C), Degree of Saturation (DS), Delay (D), Probability of Queuing (QP%).

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17

Walczak, Bartosz. "Triangle-Free Geometric Intersection Graphs with No Large Independent Sets." Discrete & Computational Geometry 53, no. 1 (November 12, 2014): 221–25. http://dx.doi.org/10.1007/s00454-014-9645-y.

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18

BEREG, SERGEY, ADRIAN DUMITRESCU, and MINGHUI JIANG. "MAXIMUM AREA INDEPENDENT SETS IN DISK INTERSECTION GRAPHS." International Journal of Computational Geometry & Applications 20, no. 02 (April 2010): 105–18. http://dx.doi.org/10.1142/s0218195910003220.

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Maximum Independent Set (MIS) and its relative Maximum Weight Independent Set (MWIS) are well-known problems in combinatorial optimization; they are NP-hard even in the geometric setting of unit disk graphs. In this paper, we study the Maximum Area Independent Set (MAIS) problem, a natural restricted version of MWIS in disk intersection graphs where the weight equals the disk area. We obtain: (i) Quantitative bounds on the maximum total area of an independent set relative to the union area; (ii) Practical constant-ratio approximation algorithms for finding an independent set with a large total area relative to the union area.

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19

Fox, Jacob, and János Pach. "Coloring Kk-free intersection graphs of geometric objects in the plane." European Journal of Combinatorics 33, no. 5 (July 2012): 853–66. http://dx.doi.org/10.1016/j.ejc.2011.09.021.

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20

Iyer, Srikanth K., and D. Yogeshwaran. "Percolation and Connectivity in AB Random Geometric Graphs." Advances in Applied Probability 44, no. 1 (March 2012): 21–41. http://dx.doi.org/10.1239/aap/1331216643.

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Given two independent Poisson point processes Φ(1), Φ(2) in , the AB Poisson Boolean model is the graph with the points of Φ(1) as vertices and with edges between any pair of points for which the intersection of balls of radius 2r centered at these points contains at least one point of Φ(2). This is a generalization of the AB percolation model on discrete lattices. We show the existence of percolation for all d ≥ 2 and derive bounds for a critical intensity. We also provide a characterization for this critical intensity when d = 2. To study the connectivity problem, we consider independent Poisson point processes of intensities n and τn in the unit cube. The AB random geometric graph is defined as above but with balls of radius r. We derive a weak law result for the largest nearest-neighbor distance and almost-sure asymptotic bounds for the connectivity threshold.

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21

Iyer, Srikanth K., and D. Yogeshwaran. "Percolation and Connectivity in AB Random Geometric Graphs." Advances in Applied Probability 44, no. 01 (March 2012): 21–41. http://dx.doi.org/10.1017/s0001867800005437.

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Given two independent Poisson point processes Φ(1), Φ(2) in , the AB Poisson Boolean model is the graph with the points of Φ(1) as vertices and with edges between any pair of points for which the intersection of balls of radius 2r centered at these points contains at least one point of Φ(2). This is a generalization of the AB percolation model on discrete lattices. We show the existence of percolation for all d ≥ 2 and derive bounds for a critical intensity. We also provide a characterization for this critical intensity when d = 2. To study the connectivity problem, we consider independent Poisson point processes of intensities n and τn in the unit cube. The AB random geometric graph is defined as above but with balls of radius r. We derive a weak law result for the largest nearest-neighbor distance and almost-sure asymptotic bounds for the connectivity threshold.

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22

De, Minati, Subhas C. Nandy, and Sasanka Roy. "In-place algorithms for computing a largest clique in geometric intersection graphs." Discrete Applied Mathematics 178 (December 2014): 58–70. http://dx.doi.org/10.1016/j.dam.2014.06.025.

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23

Böltz, Lucas, and Hannes Frey. "Automatically Testing Containedness between Geometric Graph Classes Defined by Inclusion, Exclusion, and Transfer Axioms under Simple Transformations." Information 13, no. 12 (December 12, 2022): 578. http://dx.doi.org/10.3390/info13120578.

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We study classes of geometric graphs, which all correspond to the following structural characteristic. For each instance of a vertex set drawn from a universe of possible vertices, each pair of vertices is either required to be connected, forbidden to be connected, or existence or non-existence of an edge is undetermined. The conditions which require or forbid edges are universally quantified predicates defined over the vertex pair, and optionally over existence or non-existence of another edge originating at the vertex pair. We consider further a set of simple graph transformations, where the existence of an edge between two vertices is logically determined by the existence or non-existence of directed edges between both vertices in the original graph. We derive and prove the correctness of a logical expression, which is a necessary and sufficient condition for containedness relations between graph classes that are described this way. We apply the expression on classes of geometric graphs, which are used as theoretical wireless network graph models. The models are constructed from three base class types and intersection combinations of them, with some considered directly and some considered as symmetrized variants using two of the simple graph transformations. Our study then goes systematically over all possible graph classes resulting from those base classes and all possible simple graph transformations. We derive automatically containedness relations between those graph classes. Moreover, in those cases where containedness does not hold, we provide automatically derived counter examples.

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24

de Berg, Mark, Hans L. Bodlaender, Sándor Kisfaludi-Bak, Dániel Marx, and Tom C. van der Zanden. "A Framework for Exponential-Time-Hypothesis--Tight Algorithms and Lower Bounds in Geometric Intersection Graphs." SIAM Journal on Computing 49, no. 6 (January 2020): 1291–331. http://dx.doi.org/10.1137/20m1320870.

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25

Soroko, Ignat. "Realizable ranks of joins and intersections of subgroups in free groups." International Journal of Algebra and Computation 30, no. 03 (December 9, 2019): 625–66. http://dx.doi.org/10.1142/s0218196720500149.

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The famous Hanna Neumann Conjecture (now the Friedman–Mineyev theorem) gives an upper bound for the ranks of the intersection of arbitrary subgroups [Formula: see text] and [Formula: see text] of a non-abelian free group. It is an interesting question to “quantify” this bound with respect to the rank of [Formula: see text], the subgroup generated by [Formula: see text] and [Formula: see text]. We describe a set of realizable values [Formula: see text] for arbitrary [Formula: see text], [Formula: see text], and conjecture that this locus is complete. We study the combinatorial structure of the topological pushout of the core graphs for [Formula: see text] and [Formula: see text] with the help of graphs introduced by Dicks in the context of his Amalgamated Graph Conjecture. This allows us to show that certain conditions on ranks of [Formula: see text], [Formula: see text] are not realizable, thus resolving the remaining open case [Formula: see text] of Guzman’s “Group-Theoretic Conjecture” in the affirmative. This in turn implies the validity of the corresponding “Geometric Conjecture” on hyperbolic 3-manifolds with a 6-free fundamental group. Finally, we prove the main conjecture describing the locus of realizable values for the case when [Formula: see text].

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26

Shvets, S. V., and V. P. Astakhov. "Effect of Insert Angles on Cutting Tool Geometry." Journal of Engineering Sciences 7, no. 2 (2020): A1—A6. http://dx.doi.org/10.21272/jes.2020.7(2).a1.

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An analysis of publications has shown that mechanically clamped indexable inserts are predominantly used in modern tool manufacturing. Each insert has its shape and geometry in the tool coordinate system. The static system’s required geometry is achieved by the tilting of the insert pocket in the radial and axial directions. Therefore, it is of great importance in the tool design to know the relationships between the insert’s geometry parameters in the tool coordinate system where the geometry paraments of the insert are defined and working geometry parameters of the tool defined in the static coordinate system. The paper presents the developed methodology for determining the insert pocket base surface position to ensure the required values of the tool geometry parameters of the selected indexable insert in the static coordinate system. The graphs of the dependence of each of the angles of the insert geometry on the angles of rotation of this insert in the front and profile planes are presented as the level lines for practical use. Using these graphs, one can optimize all geometric insertion parameters in the static coordinate system. The model of the calculations of the mechanism of the insert clamping by a screw is developed. The basic size and tolerance of the output link determine the distance from the intersection line of the base surfaces to the thread axis on the pocket and the minimum amount of the screw stroke on the insert clamping in the pocket. Keywords: indexable insert, cutting tool, coordinate system, base surfaces, geometric parameters.

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27

Wang, Wei, Yi Zhang, Gengyu Ge, Huan Yang, and Yue Wang. "A New Approach toward Corner Detection for Use in Point Cloud Registration." Remote Sensing 15, no. 13 (July 1, 2023): 3375. http://dx.doi.org/10.3390/rs15133375.

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For this study, a new point cloud alignment method is proposed for extracting corner points and aligning them at the geometric level. It can align point clouds that have low overlap and is more robust to outliers and noise. First, planes are extracted from the raw point cloud, and the corner points are defined as the intersection of three planes. Next, graphs are constructed for subsequent point cloud registration by treating corners as vertices and sharing planes as edges. The graph-matching algorithm is then applied to determine correspondence. Finally, point clouds are registered by aligning the corresponding corner points. The proposed method was investigated by utilizing pertinent metrics on datasets with differing overlap. The results demonstrate that the proposed method can align point clouds that have low overlap, yielding an RMSE of about 0.05 cm for datasets with 90% overlap and about 0.2 cm when there is only about 10% overlap. In this situation, the other methods failed to align point clouds. In terms of time consumption, the proposed method can process a point cloud comprising 104 points in 4 s when there is high overlap. When there is low overlap, it can also process a point cloud comprising 106 points in 10 s. The contributions of this study are the definition and extraction of corner points at the geometric level, followed by the use of these corner points to register point clouds. This approach can be directly used for low-precision applications and, in addition, for coarse registration in high-precision applications.

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28

Bang, Karl-L., Gunnar Lindberg, and Gandhi Harahap. "Development of Guidelines Based on Life-Cycle Cost To Replace Level-of-Service Concept in Capacity Analysis." Transportation Research Record: Journal of the Transportation Research Board 1572, no. 1 (January 1997): 10–17. http://dx.doi.org/10.3141/1572-02.

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Highway capacity manuals (HCMs) from developed countries cannot be successfully applied in Indonesia because of significant differences in driver behavior, traffic composition, and level of roadside activities. The Indonesian HCM project (IHCM) was therefore started in 1991 and has resulted in a manual and software for urban traffic facilities, inter-urban roads, and motorways. The IHCM also includes traffic engineering guidelines for selection of appropriate road and intersection type and design for new construction or improvements based on life-cycle cost (LCC) analysis. The LCC model includes all relevant road-user costs (those for vehicle operations, time, accidents, and emissions) and road-producer costs (those for land acquisition, road construction, and road maintenance and operation). The LCC results identify the design alternative that has the lowest total cost and thus is the most economical for any given traffic flow. The IHCM guidelines also include traffic-performance graphs for standard road types and intersections as a function of traffic flow as well as information about accident rates and the impact of geometric design changes on safety. The guidelines assist the user with preliminary selection of the design before starting the detailed analysis. They can also be used on their own as a shortcut for planning purposes and therefore serve as a replacement for the level-of-service concept applied in the U.S. HCM.

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29

Cayron, Cyril. "The transformation matrices (distortion, orientation, correspondence), their continuous forms and their variants." Acta Crystallographica Section A Foundations and Advances 75, no. 3 (April 10, 2019): 411–37. http://dx.doi.org/10.1107/s205327331900038x.

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The crystallography of displacive/martensitic phase transformations can be described with three types of matrix: the lattice distortion matrix, the orientation relationship matrix and the correspondence matrix. Given here are some formulae to express them in crystallographic, orthonormal and reciprocal bases, and an explanation is offered of how to deduce the matrices of inverse transformation. In the case of the hard-sphere assumption, a continuous form of distortion matrix can be determined, and its derivative is identified to the velocity gradient used in continuum mechanics. The distortion, orientation and correspondence variants are determined by coset decomposition with intersection groups that depend on the point groups of the phases and on the type of transformation matrix. The stretch variants required in the phenomenological theory of martensitic transformation should be distinguished from the correspondence variants. The orientation and correspondence variants are also different; they are defined from the geometric symmetries and algebraic symmetries, respectively. The concept of orientation (ir)reversibility during thermal cycling is briefly and partially treated by generalizing the orientation variants with n-cosets and graphs. Some simple examples are given to show that there is no general relation between the numbers of distortion, orientation and correspondence variants, and to illustrate the concept of orientation variants formed by thermal cycling.

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30

Patrikalakis, N. M., and P. V. Prakash. "Surface Intersections for Geometric Modeling." Journal of Mechanical Design 112, no. 1 (March 1, 1990): 100–107. http://dx.doi.org/10.1115/1.2912565.

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Evaluation of planar algebraic curves arises in the context of intersections of algebraic surfaces with piecewise continuous rational polynomial parametric surface patches useful in geometric modeling. We address a method of evaluating these curves of intersection that combines the advantageous features of analytic representation of the governing equation of the algebraic curve in the Bernstein basis within a rectangular domain, adaptive subdivision and polyhedral faceting techniques, and the computation of turning and singular points, to provide the basis for a reliable and efficient solution procedure. Using turning and singular points, the intersection problem can be partitioned into subdomains that can be processed independently and which involve intersection segments that can be traced with faceting methods. This partitioning and the tracing of individual segments is carried out using an adaptive subdivision algorithm for Bezier/B-spline surfaces followed by Newton correction of the approximation. The method has been successfully tested in tracing complex algebraic curves and in solving actual intersection problems with diverse features.

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31

Bollobás, Béla, and Alex Scott. "Intersections of graphs." Journal of Graph Theory 66, no. 4 (February 9, 2011): 261–82. http://dx.doi.org/10.1002/jgt.20489.

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32

Deng, Zhiyun, Yanjun Shi, Qiaomei Han, Lingling Lv, and Weiming Shen. "A Conflict Duration Graph-Based Coordination Method for Connected and Automated Vehicles at Signal-Free Intersections." Applied Sciences 10, no. 18 (September 8, 2020): 6223. http://dx.doi.org/10.3390/app10186223.

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Previous studies on Connected and Automated Vehicles (CAVs) demonstrated the potential to coordinate the behaviors of multiple connected vehicles for traffic improvements. In this paper, we first propose a Conflict Duration Graph-based (CDG-based) coordination framework to resolve collisions and improve the traffic capacity of signal-free intersections. Secondly, a Speed Control-based Intersection Coordination Model (SICM) is developed to identify complex constraints in multi-vehicle collision scenarios. Thirdly, a geometric Translation-based Intersection Coordination Algorithm (TICA) is proposed to calculate the ideal location of time blocks in CDGs and then obtain the near-optimal design speed in the form of combinatorial optimization. Twelve groups of test scenarios with different traffic volumes were designed and tested on a MATLAB-based simulation platform. Simulation results showed that the proposed method can resolve all the collisions and instruct the vehicles to pass signal-free intersections collaboratively without stopping in low to medium level of congestion.

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33

Gavril, Fǎnicǎ. "Intersection graphs of proper subtrees of unicyclic graphs." Journal of Graph Theory 18, no. 6 (October 1994): 615–27. http://dx.doi.org/10.1002/jgt.3190180609.

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34

Chmutov, Sergei, and Sergei Lando. "Mutant knots and intersection graphs." Algebraic & Geometric Topology 7, no. 3 (December 17, 2007): 1579–98. http://dx.doi.org/10.2140/agt.2007.7.1579.

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35

Cai, Leizhen, Derek Corneil, and Andrzej Proskurowski. "A generalization of line graphs: (X, Y)-intersection graphs." Journal of Graph Theory 21, no. 3 (March 1996): 267–87. http://dx.doi.org/10.1002/(sici)1097-0118(199603)21:3<267::aid-jgt2>3.0.co;2-o.

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36

Jukna, Stasys. "On set intersection representations of graphs." Journal of Graph Theory 61, no. 1 (May 2009): 55–75. http://dx.doi.org/10.1002/jgt.20367.

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37

Chen, G., and R. H. Schelp. "Hamiltonian graphs with neighborhood intersections." Journal of Graph Theory 18, no. 5 (August 1994): 497–513. http://dx.doi.org/10.1002/jgt.3190180508.

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38

Giovanella, A., P. E. Bradley, and S. Wursthorn. "DETECTION AND EVALUATION OF TOPOLOGICAL CONSISTENCY IN CITYGML DATASETS." ISPRS Annals of Photogrammetry, Remote Sensing and Spatial Information Sciences IV-4 (September 19, 2018): 59–66. http://dx.doi.org/10.5194/isprs-annals-iv-4-59-2018.

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<p><strong>Abstract.</strong> The topological consistency of Boundary-Representation models, meaning here that the incidence graph is homeomorphic with the underlying topology of geographical data, is checked for several CityGML datasets, and a first classification of topological inconsistencies is performed. The analysis is carried out on a spatial database system into which the datasets have been imported. It is found that real-world datasets contain many topologically inconsistent pairs of intersecting polygons. Also data satisfying the ISO/OGC standards can still be topologically inconsistent. In the case when the intersection is a point, topological inconsistency occurs because a vertex lies on a line segment. However, the most frequent topological inconsistencies seem to arise when the intersection of two polygons is a line segment. Consequently, topological queries in present CityGML data cannot rely on the incidence graph only, but must always make costly geometric computations if correct results are to be expected.</p>

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39

Golubyatnikov, V. P., A. A. Akinshin, N. B. Ayupova, and L. S. Minushkina. "Stratifications and foliations in phase portraits of gene network models." Vavilov Journal of Genetics and Breeding 26, no. 8 (January 4, 2023): 758–64. http://dx.doi.org/10.18699/vjgb-22-91.

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Periodic processes of gene network functioning are described with good precision by periodic trajectories (limit cycles) of multidimensional systems of kinetic-type differential equations. In the literature, such systems are often called dynamical, they are composed according to schemes of positive and negative feedback between components of these networks. The variables in these equations describe concentrations of these components as functions of time. In the preparation of numerical experiments with such mathematical models, it is useful to start with studies of qualitative behavior of ensembles of trajectories of the corresponding dynamical systems, in particular, to estimate the highest likelihood domain of the initial data, to solve inverse problems of parameter identification, to list the equilibrium points and their characteristics, to localize cycles in the phase portraits, to construct stratification of the phase portraits to subdomains with different qualities of trajectory behavior, etc. Such an à priori geometric analysis of the dynamical systems is quite analogous to the basic section “Investigation of functions and plot of their graphs” of Calculus, where the methods of qualitative studies of shapes of curves determined by equations are exposed. In the present paper, we construct ensembles of trajectories in phase portraits of some dynamical systems. These ensembles are 2-dimensional surfaces invariant with respect to shifts along the trajectories. This is analogous to classical construction in analytic mechanics, i. e. the level surfaces of motion integrals (energy, kinetic moment, etc.). Such surfaces compose foliations in phase portraits of dynamical systems of Hamiltonian mechanics. In contrast with this classical mechanical case, the foliations considered in this paper have singularities: all their leaves have a non-empty intersection, they contain limit cycles on their boundaries. Description of the phase portraits of these systems at the level of their stratifications, and that of ensembles of trajectories allows one to construct more realistic gene network models on the basis of methods of statistical physics and the theory of stochastic differential equations.

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40

Eaton, Nancy, and David A. Grable. "Set intersection representations for almost all graphs." Journal of Graph Theory 23, no. 3 (November 1996): 309–20. http://dx.doi.org/10.1002/(sici)1097-0118(199611)23:3<309::aid-jgt11>3.0.co;2-9.

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Tanenbaum, Paul J. "Simultaneous intersection representation of pairs of graphs." Journal of Graph Theory 32, no. 2 (October 1999): 171–90. http://dx.doi.org/10.1002/(sici)1097-0118(199910)32:2<171::aid-jgt7>3.0.co;2-n.

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Pach, János, and Micha Sharir. "On planar intersection graphs with forbidden subgraphs." Journal of Graph Theory 59, no. 3 (November 2008): 205–14. http://dx.doi.org/10.1002/jgt.20332.

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Fachini, Emanuela, and János Körner. "Cross-intersecting couples of graphs." Journal of Graph Theory 56, no. 2 (2007): 105–12. http://dx.doi.org/10.1002/jgt.20252.

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Pach, János, Rom Pinchasi, Gábor Tardos, and Géza Tóth. "Geometric graphs with no self-intersecting path of length three." European Journal of Combinatorics 25, no. 6 (August 2004): 793–811. http://dx.doi.org/10.1016/j.ejc.2003.09.019.

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Fys, Mykhailo, Volodymyr Litynskyi, Anatolii Vivat, and Svyatoslav Litynskyi. "GEODESY, CARTOGRAPHY AND AERIAL PHOTOGRAPHY." GEODESY, CARTOGRAPHY AND AERIAL PHOTOGRAPHY 94, 2021, no. 94 (December 28, 2021): 20–28. http://dx.doi.org/10.23939/istcgcap2021.94.020.

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The aim. The study of formulas determination of the point coordinates by the inverse linear-angular intersection method. Previously, we investigated the possibility of using electronic total stations to control the geometric parameters of industrial buildings. The applied application of electronic total stations for high-precision measurements has been investigated as well. [Vivat, 2018]. The formula for optimal use of the device with certain accuracy characteristics relative to the measured basis is analytically proved and derived [Litynskyi, 2014]. Measurements on the basis of the II category are performed and theoretical calculations are confirmed. The possibility of achieving high accuracy in determining the segment by the method of linear-angular measurements is shown [Litynsky, 2015]. The influence of the angle value on the accuracy of determining the coordinates by the sine theorem is investigated and the possibility of optimizing the determination of coordinates by the method of inverse linear-angular serif by the formulas of cosines and sines is investigated [Litynskyi, 2019]. Method. Establishing a mathematical interconnection between measured values (distances and angles) with the required (flat coordinates of a point), differentiation and finding the minima of functions. Results.There were five formulas selected, of which six combinations had been created to calculate the increments of coordinates and to estimate their accuracy. Numerical experiments show that neither method has a significant advantage, which is supported by the results presented in the graphs and tables. It is worth noting one feature of the second method - in which it is possible to determine the increments of coordinates with an accuracy that exceeds the accuracy of measuring the sides. The possibility of optimizing the coordinate increments determination due to the choice of calculation formulas is considered. The possibility of increasing the accuracy of determination of the coordinates increments using different calculation formulas is researched. Consequently, it is suggested to optimize the choice of calculation formulas depending on the position of the desired point. The results of these studies can be used to create electronic total station or laser tracker application software in order to improve the accuracy of coordinate determination.

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Menke, B., T. Zamfirescu, and C. Zamfirescu. "Intersections of longest cycles in grid graphs." Journal of Graph Theory 25, no. 1 (May 1997): 37–52. http://dx.doi.org/10.1002/(sici)1097-0118(199705)25:1<37::aid-jgt2>3.0.co;2-j.

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BOSE, PROSENJIT, HAZEL EVERETT, and STEPHEN WISMATH. "PROPERTIES OF ARRANGEMENT GRAPHS." International Journal of Computational Geometry & Applications 13, no. 06 (December 2003): 447–62. http://dx.doi.org/10.1142/s0218195903001281.

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An arrangement graph G is the abstract graph obtained from an arrangement of lines L, in general position by associating vertices of G with the intersection points of L, and the edges of G with the line segments joining the intersection points of L. A simple polygon (respectively path) of n sides in general position, induces a set of n lines by extension of the line segments into lines. The main results of this paper are: • Given a graph G, it is NP-Hard to determine if G is the arrangement graph of some set of lines. • There are non-Hamiltonian arrangement graphs for arrangements of six lines and for odd values of n>6 lines. • All arrangements of n lines contain a subarrangement of size [Formula: see text] with an inducing polygon. • All arrangements on n lines contain an inducing path consisting of n line segments. A Java applet implementing the algorithm for determining such a path is also provided. • All arrangements on n hyperplanes in Rd contain a simple inducing polygonal cycle of size n.

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Watanabe, Yohsuke. "Distances and Intersections of Curves." International Mathematics Research Notices 2020, no. 23 (November 14, 2018): 9674–93. http://dx.doi.org/10.1093/imrn/rny265.

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Abstract We obtain a coarse relationship between geometric intersection numbers of curves and the sum of their subsurface projection distances with explicit quasi-constants. By using this relationship, we study intersection numbers of curves contained in geodesics in the curve graph. Furthermore, we generalize a well-known result on intersection number growth of curves under iteration of Dehn twists and multitwists for all kinds of pure mapping classes.

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Paliy, N. "The Use of Descriptive Geometry Methods for Graphical Study of the Parameters of the Cutting Part of a Spiral Drill." Geometry & Graphics 11, no. 2 (August 21, 2023): 39–46. http://dx.doi.org/10.12737/2308-4898-2023-11-2-39-46.

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The article presents examples of practical application of descriptive geometry methods for solving engineering problems of determining and studying the parameters of the cutting part of a spiral drill. The necessity of graphical constructions is noted, which is due to the fact that it is impossible to measure the rear angle by a direct measurement method using a universal measuring instrument due to the complex geometric shape of the measuring object. The analysis of the geometric shape of the working part of the spiral drill, which includes the main rear conical surface of the drill tooth, the front screw surface of the drill groove, the cylindrical surface of the tooth back, the main and transverse cutting edges are presented. Using the double contact theorem, a projection of the tooth profile of a spiral drill is constructed on a plane parallel to the drill axis with a given profile projection on a plane perpendicular to the drill axis. The tooth profile is determined by the lines of intersection of the conical surface with the cylindrical surface and with the helical surface of the drill groove. The shape of the drill groove is considered to be set. Projections of the transverse cutting edge of the drill are constructed as lines of intersection of two conical surfaces with intersecting axes. The problem is solved using a bundle of auxiliary planes. The tendency of an increase in the angle of inclination of the transverse cutting edge to the drill axis with a decrease in the distance between the intersecting axes of conical surfaces is shown. Graphically, at an arbitrary point of the drill blade, the value of one of the main geometric parameters of the drill - the rear angle is determined. It is shown that the closer the point of the blade is to the center of the drill, the greater the value of the rear angle. The constructions are made in the 2D Compass-Graph design system. The paper used alternative methods for solving geometric problems, which, unlike traditional ones, provide a simpler and more visual graphical solution.

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Petitjean, Sylvain. "A Computational Geometric Approach to Visual Hulls." International Journal of Computational Geometry & Applications 08, no. 04 (August 1998): 407–36. http://dx.doi.org/10.1142/s0218195998000229.

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Recognizing 3D objects from their 2D silhouettes is a popular topic in computer vision. Object reconstruction can be performed using the volume intersection approach. The visual hull of an object is the best approximation of an object that can be obtained by volume intersection. From the point of view of recognition from silhouettes, the visual hull can not be distinguished from the original object. In this paper, we present efficient algorithms for computing visual hulls. We start with the case of planar figures (polygons and curved objects) and base our approach on an efficient algorithm for computing the visibility graph of planar figures. We present and tackle many topics related to the query of visual hulls and to the recognition of objects equal to their visual hulls. We then move on to the 3-dimensional case and give a flavor of how it may be approached.

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