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Journal articles on the topic 'Graph drawing'

Author: Grafiati

Published: 4 June 2021

Last updated: 20 February 2023

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1

EADES, PETER, XUEMIN LIN, and ROBERTO TAMASSIA. "AN ALGORITHM FOR DRAWING A HIERARCHICAL GRAPH." International Journal of Computational Geometry & Applications 06, no. 02 (June 1996): 145–55. http://dx.doi.org/10.1142/s0218195996000101.

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Hierarchical graphs appear in several graph drawing applications, where nodes are assigned layers for semantic reasons. More importantly, general methods for drawing directed graphs usually begin by transforming the input digraph into a hierarchical graph, then applying a hierarchical graph drawing algorithm. This paper introduces the Degree Weighted Barycentre (DWB) algorithm for drawing hierarchical graphs. We show that drawings output by DWB satisfy several important aesthetic criteria: under certain connectivity conditions, they are planar, convex, and symmetric whenever such drawings are possible. The algorithm can be implemented as a simple Gauss — Seidel iteration.

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2

BRIDGEMAN, STINA, ASHIM GARG, and ROBERTO TAMASSIA. "A GRAPH DRAWING AND TRANSLATION SERVICE ON THE WORLD WIDE WEB." International Journal of Computational Geometry & Applications 09, no. 04n05 (August 1999): 419–46. http://dx.doi.org/10.1142/s021819599900025x.

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Both practitioners and researchers can take better advantage of the latest developments in graph drawing if implementations of graph drawing algorithms are made available on the WWW. We envision a graph drawing and translation service for the WWW with dual objectives: drawing user-specified graphs, and translating graph-descriptions and graph drawings from one format to another. As a first step toward realizing this vision, we have developed a prototype service which is available at .

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3

Iqbal Hossain, Md, and Md Saidur Rahman. "Straight-line monotone grid drawings of series–parallel graphs." Discrete Mathematics, Algorithms and Applications 07, no. 02 (May 25, 2015): 1550007. http://dx.doi.org/10.1142/s179383091550007x.

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A monotone drawing of a planar graph G is a planar straight-line drawing of G where a monotone path exists between every pair of vertices of G in some direction. Recently monotone drawings of graphs have been discovered as a new standard for visualizing graphs. In this paper we study monotone drawings of series–parallel graphs in a variable embedding setting. We show that a series–parallel graph of n vertices has a straight-line planar monotone drawing on a grid of size O(n) × O(n2) and such a drawing can be found in linear time.

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4

Kang, Ming-Hsuan, and Jing-Wen Gu. "Toroidal Spectral Drawing." Axioms 11, no. 3 (March 16, 2022): 137. http://dx.doi.org/10.3390/axioms11030137.

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We give a deterministic drawing algorithm to draw a graph onto a torus, which is based on the usual spectral drawing algorithm. For most of the well-known toroidal vertex-transitive graphs, the result drawings give an embedding of the graphs onto the torus.

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5

BIEDL, THERESE C., BRENDAN P. MADDEN, and IOANNIS G. TOLLIS. "THE THREE-PHASE METHOD: A UNIFIED APPROACH TO ORTHOGONAL GRAPH DRAWING." International Journal of Computational Geometry & Applications 10, no. 06 (December 2000): 553–80. http://dx.doi.org/10.1142/s0218195900000310.

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In this paper, we study orthogonal graph drawings from a practical point of view. Most previously existing algorithms restricted the attention to graphs of maximum degree four. Here we study orthogonal drawing algorithms that work for any input graph, and discuss different models for such drawings. Then we introduce the three-phase method, a generic technique to create high-degree orthogonal drawings. This approach simplifies the description and implementation of orthogonal graph drawing, and can be applied to global as well as interactive and incremental settings.

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6

Bertolazzi, P., G. Di Battista, and G. Liotta. "Parametric graph drawing." IEEE Transactions on Software Engineering 21, no. 8 (1995): 662–73. http://dx.doi.org/10.1109/32.403790.

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7

DEHKORDI, HOOMAN REISI, and PETER EADES. "EVERY OUTER-1-PLANE GRAPH HAS A RIGHT ANGLE CROSSING DRAWING." International Journal of Computational Geometry & Applications 22, no. 06 (December 2012): 543–57. http://dx.doi.org/10.1142/s021819591250015x.

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There is strong empirical evidence that human perception of a graph drawing is negatively correlated with the number of edge crossings. However, recent experiments show that one can reduce the negative effect by ensuring that the edges that cross do so at large angles. These experiments have motivated a number of mathematical and algorithmic studies of “right angle crossing (RAC)” drawings of graphs, where the edges cross each other perpendicularly. In this paper we give an algorithm for constructing RAC drawings of “outer-1-plane” graphs, that is, topological graphs in which each vertex appears on the outer face, and each edge crosses at most one other edge. The drawing algorithm preserves the embedding of the input graph. This is one of the few algorithms available to construct RAC drawings.

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8

MIURA, KAZUYUKI, SHIN-ICHI NAKANO, and TAKAO NISHIZEKI. "CONVEX GRID DRAWINGS OF FOUR-CONNECTED PLANE GRAPHS." International Journal of Foundations of Computer Science 17, no. 05 (October 2006): 1031–60. http://dx.doi.org/10.1142/s012905410600425x.

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A convex grid drawing of a plane graph G is a drawing of G on the plane such that all vertices of G are put on grid points, all edges are drawn as straight-line segments without any edge-intersection, and every face boundary is a convex polygon. In this paper we give a linear-time algorithm for finding a convex grid drawing of every 4-connected plane graph G with four or more vertices on the outer face. The size of the drawing satisfies W + H ≤ n - 1, where n is the number of vertices of G, W is the width and H is the height of the grid drawing. Thus the area W · H is at most ⌈(n - 1)/2⌉ · ⌊(n - 1)/2⌋. Our bounds on the sizes are optimal in a sense that there exist an infinite number of 4-connected plane graphs whose convex drawings need grids such that W + H = n - 1 and W · H = ⌈(n - 1)/2⌉ · ⌊(n - 1)/2⌋.

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9

Liotta, Giuseppe, and Henk Meijer. "Advances in graph drawing: The 11th International Symposium on Graph Drawing." Discrete Applied Mathematics 155, no. 9 (May 2007): 1077. http://dx.doi.org/10.1016/j.dam.2006.10.002.

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10

A. Antony mary, A., A. Amutha, and M. S. Franklin Thamil Selvi. "A Study on Slope Number of Certain Classes of Bipartite Graphs." International Journal of Engineering & Technology 7, no. 4.10 (October 2, 2018): 440. http://dx.doi.org/10.14419/ijet.v7i4.10.21036.

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Graph drawing is the most important area of mathematics and computer science which combines methods from geometric graph theory and information visualization. Generally, graphs are represented to explore some intellectual ideas. Graph drawing is the familiar concept of graph theory. It has many quality measures and one among them is the slope number. Slope number problem is an optimization problem and is NP-hard to determine the slope number of any arbitrary graph. In the present paper, the investigation on slope number of bipartite graph is studied elaborately. Since the bipartite graphs creates one of the most intensively investigated classes of graphs, we consider few classes of graphs and discussed structural behavior of such graphs.

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11

McCreary, C. L., R. O. Chapman, and F. S. Shieh. "Using graph parsing for automatic graph drawing." IEEE Transactions on Systems, Man, and Cybernetics - Part A: Systems and Humans 28, no. 5 (1998): 545–61. http://dx.doi.org/10.1109/3468.709599.

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12

Friedrich, Carsten, and Peter Eades. "Graph Drawing in Motion." Journal of Graph Algorithms and Applications 6, no. 3 (2002): 353–70. http://dx.doi.org/10.7155/jgaa.00057.

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13

Tantau, Till. "Graph Drawing in TikZ." Journal of Graph Algorithms and Applications 17, no. 4 (2013): 495–513. http://dx.doi.org/10.7155/jgaa.00301.

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14

Frishman, Y., and A. Tal. "Online Dynamic Graph Drawing." IEEE Transactions on Visualization and Computer Graphics 14, no. 4 (July 2008): 727–40. http://dx.doi.org/10.1109/tvcg.2008.11.

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15

Cohen, Robert F. "Dynamic graph drawing (abstract)." ACM SIGACT News 24, no. 1 (January 15, 1993): 60. http://dx.doi.org/10.1145/152992.153005.

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16

Papakostas, A., and I. G. Tollis. "Interactive orthogonal graph drawing." IEEE Transactions on Computers 47, no. 11 (1998): 1297–309. http://dx.doi.org/10.1109/12.736444.

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17

Cohen, R. F., P. Eades, Tao Lin, and F. Ruskey. "Three-dimensional graph drawing." Algorithmica 17, no. 2 (February 1997): 199–208. http://dx.doi.org/10.1007/bf02522826.

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18

Jing-wei, Huang, and Wei Wen-fang. "Evolutionary graph drawing algorithms." Wuhan University Journal of Natural Sciences 8, no. 1 (March 2003): 212–16. http://dx.doi.org/10.1007/bf02899481.

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19

DUNCAN, CHRISTIAN A., ALON EFRAT, STEPHEN KOBOUROV, and CAROLA WENK. "DRAWING WITH FAT EDGES." International Journal of Foundations of Computer Science 17, no. 05 (October 2006): 1143–63. http://dx.doi.org/10.1142/s0129054106004315.

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Traditionally, graph drawing algorithms represent vertices as circles and edges as curves connecting the vertices. We introduce the problem of drawing with "fat" edges, i.e., with edges of variable thickness. The thickness of an edge is often used as a visualization cue, to indicate importance, or to convey some additional information. We present a model for drawing with fat edges and a corresponding efficient polynomial time algorithm that uses the model. We first focus on a restricted class of graphs that occur in VLSI wire routing and then show how to extend the algorithm to general planar graphs. We show how to convert an arbitrary wire routing into a homotopically equivalent routing that maximizes the distance between any two wires, which is a desired property in VLSI design. Among such, we obtain the routing with minimum total wire length. A homotopically equivalent routing that maximizes the distance between any two wires yields a graph drawing which maximizes edge thickness. Our algorithm does not require unit edge thickness but can be applied as well in the presence of different edge weights.

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20

Brückner, Guido, Nadine Krisam, and Tamara Mchedlidze. "Level-Planar Drawings with Few Slopes." Algorithmica 84, no. 1 (November 19, 2021): 176–96. http://dx.doi.org/10.1007/s00453-021-00884-x.

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AbstractWe introduce and study level-planar straight-line drawings with a fixed number $$\lambda $$ λ of slopes. For proper level graphs (all edges connect vertices of adjacent levels), we give an $$O(n \log ^2 n / \log \log n)$$ O ( n log 2 n / log log n ) -time algorithm that either finds such a drawing or determines that no such drawing exists. Moreover, we consider the partial drawing extension problem, where we seek to extend an immutable drawing of a subgraph to a drawing of the whole graph, and the simultaneous drawing problem, which asks about the existence of drawings of two graphs whose restrictions to their shared subgraph coincide. We present $$O(n^{4/3} \log n)$$ O ( n 4 / 3 log n ) -time and $$O(\lambda n^{10/3} \log n)$$ O ( λ n 10 / 3 log n ) -time algorithms for these respective problems on proper level-planar graphs. We complement these positive results by showing that testing whether non-proper level graphs admit level-planar drawings with $$\lambda $$ λ slopes is -hard even in restricted cases.

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21

Czap, Július, and Peter Sugerek. "Drawing graph joins in the plane with restrictions on crossings." Filomat 31, no. 2 (2017): 363–70. http://dx.doi.org/10.2298/fil1702363c.

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A graph is called 1-planar if it can be drawn in the plane so that each of its edges is crossed by at most one other edge. In 2014, Zhang showed that the set of all 1-planar graphs can be decomposed into three classes C0,C1 and C2 with respect to the types of crossings. He proved that every n-vertex 1-planar graph of class C1 has a C1-drawing with at most 3/5n-6/5 crossings. Consequently, every n-vertex 1-planar graph of class C1 has at most 18/5n ? 36/5 edges. In this paper we prove a stronger result. We show that every C1-drawing of a 1-planar graph has at most 3/5n ? 6/5 crossings. Next we present a construction of n-vertex 1-planar graphs of class C1 with 18/5n ? 36/5 edges. Finally, we present the decomposition of 1-planar join products.

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22

Koch, Sebastian. "About Supergraphs. Part II." Formalized Mathematics 26, no. 2 (July 1, 2018): 125–40. http://dx.doi.org/10.2478/forma-2018-0010.

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Summary In the previous article [5] supergraphs and several specializations to formalize the process of drawing graphs were introduced. In this paper another such operation is formalized in Mizar [1], [2]: drawing a vertex and then immediately drawing edges connecting this vertex with a subset of the other vertices of the graph. In case the new vertex is joined with all vertices of a given graph G, this is known as the join of G and the trivial loopless graph K1. While the join of two graphs is known and found in standard literature (like [9], [4], [8] and [3]), the operation discribed in this article is not. Alongside the new operation a mode to reverse the directions of a subset of the edges of a graph is introduced. When all edge directions of a graph are reversed, this is commonly known as the converse of a (directed) graph.

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23

Sharma, Jitendra, and Shubhra Saxena. "Enhanced JGraphEd Drawing Framework for Graph Drawing Application." International Journal of Computer Applications 81, no. 4 (November 15, 2013): 11–16. http://dx.doi.org/10.5120/13999-2037.

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24

Shiono, Yasunori, Tadaaki Kirishima, Yoshinori Ueda, and Kensei Tsuchida. "Drawing Algorithm for Fuzzy Graphs Using the Partition Tree." Journal of Advanced Computational Intelligence and Intelligent Informatics 16, no. 5 (July 20, 2012): 641–52. http://dx.doi.org/10.20965/jaciii.2012.p0641.

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Fuzzy graphs have been used frequently and effectively as a method for sociogram analysis. A fuzzy graph has the fundamental characteristic of being able to express a variety of relationships between nodes. The drawing of fuzzy graphs has been studied in computer-aided analysis systems with human interfaces and methods using genetic algorithms. However, computer-aided analysis systems with human interfaces do not provide for automatic drawing, while methods using genetic algorithms have the defect of requiring too much execution time for finding a locally optimum solution. To overcome these defects, we propose an algorithm for drawing intelligible and comprehensive fuzzy graphs using a partition tree. This method automatically draws the fuzzy graphwith nodes arranged on the intersections of a latticed space. Since nodes are optimally arranged on the latticed intersections and put together at a nearby position in accordance with the transition of clusters according to cluster levels in the partition tree, drawing the algorithm makes fuzzy relations easier to understand through fuzzy graph representation. Moreover, fuzzy graphs can be drawn faster than by conventional methods. This paper describes the algorithm and its verification by introducing a system implementing the method for displaying fuzzy graphs. Moreover, we have carried out a case study in which a questionnaire has been administered to students, allowing us to analyze human relations quantitatively using a method based on fuzzy theory. Human relations are represented as fuzzy graphs by our algorithm and analyzed using the fuzzy graph.

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25

Jia, Jinzhang, Bin Li, Dinglin Ke, Yumo Wu, Dan Zhao, and Mingyu Wang. "Optimization of mine ventilation network feature graph." PLOS ONE 15, no. 11 (November 16, 2020): e0242011. http://dx.doi.org/10.1371/journal.pone.0242011.

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A ventilation network feature graph can directly and quantitatively represent the features of a ventilation network. To ensure the stability of airflow in a mine and improve ventilation system analysis, we propose a new algorithm to draw ventilation network feature graphs. The independent path method serves as the algorithm’s main frame, and an improved adaptive genetic algorithm is embedded so that the graph may be drawn better. A mathematical model based on the node adjacency matrix method for unidirectional circuit discrimination is constructed as the drawing algorithm may not be valid in such cases. By modifying the edge-seeking strategy, the improved depth-first search algorithm can be used to determine all of the paths in the ventilation network with unidirectional circuits, and the equivalent transformation method of network topology relations is used to draw the ventilation network feature graph. Through the analysis of the topological relation of a ventilation network, a simplified mathematical model is constructed, and network simplification technology makes the drawing concise and hierarchical. The rapid and intuitive drawing of the ventilation network feature graphs is significant for optimization of the ventilation system and day-to-day management.

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26

Akhsani, Lukmanul, and Erlin Nurhayati. "The Error in Drawing Graphic of Quadratic Function in PBL Model by Using True or False Strategy with GeoGebra-Assisted." AlphaMath : Journal of Mathematics Education 6, no. 2 (November 1, 2020): 135. http://dx.doi.org/10.30595/alphamath.v6i2.8059.

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Quadratic function is one of the important topic in mathematics. The purpose of this study is to describe the mistakes made by students when drawing a quadratic function graph. In this research, the focus of the problem is what type of mistakes made by students in drawing a graph of quadratic functions. The form of error here referred to an error in doing the exercise of drawing quadratic function graph in learning with the PBL model using a true or false strategy assisted by GeoGebra application. The conclusion of this research states that the error in drawing quadratic function graph is dominated by procedural errors. PBL model with true or false strategy with the help of GeoGebra application can be an alternative learning to overcome students' mistakes in drawing quadratic function graphs

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27

Calamoneri, Tiziana, Simone Jannelli, and Rossella Petreschi. "Experimental Comparison of Graph Drawing Algorithms for Cubic Graphs." Journal of Graph Algorithms and Applications 3, no. 2 (1999): 1–23. http://dx.doi.org/10.7155/jgaa.00013.

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28

Qu, Jianhua, Xiyu Liu, Minghe Sun, and Feng Qi. "GPU-Based Parallel Particle Swarm Optimization Methods for Graph Drawing." Discrete Dynamics in Nature and Society 2017 (2017): 1–15. http://dx.doi.org/10.1155/2017/2013673.

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Particle Swarm Optimization (PSO) is a population-based stochastic search technique for solving optimization problems, which has been proven to be effective in a wide range of applications. However, the computational efficiency on large-scale problems is still unsatisfactory. A graph drawing is a pictorial representation of the vertices and edges of a graph. Two PSO heuristic procedures, one serial and the other parallel, are developed for undirected graph drawing. Each particle corresponds to a different layout of the graph. The particle fitness is defined based on the concept of the energy in the force-directed method. The serial PSO procedure is executed on a CPU and the parallel PSO procedure is executed on a GPU. Two PSO procedures have different data structures and strategies. The performance of the proposed methods is evaluated through several different graphs. The experimental results show that the two PSO procedures are both as effective as the force-directed method, and the parallel procedure is more advantageous than the serial procedure for larger graphs.

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29

Wang, Eric Ke, and Futai Zou. "A New Graph Drawing Scheme for Social Network." Scientific World Journal 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/930314.

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With the development of social networks, people have started to use social network tools to record their life and work more and more frequently. How to analyze social networks to explore potential characteristics and trend of social events has been a hot research topic. In order to analyze it effectively, a kind of techniques called information visualization is employed to extract the potential information from the large scale of social network data and present the information briefly as visualized graphs. In the process of information visualization, graph drawing is a crucial part. In this paper, we study the graph layout algorithms and propose a new graph drawing scheme combining multilevel and single-level drawing approaches, including the graph division method based on communities and refining approach based on partitioning strategy. Besides, we compare the effectiveness of our scheme and FM3in experiments. The experiment results show that our scheme can achieve a clearer diagram and effectively extract the community structure of the social network to be applied to drawing schemes.

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30

Tollis, Ioannis G. "Graph drawing and information visualization." ACM Computing Surveys 28, no. 4es (December 1996): 19. http://dx.doi.org/10.1145/242224.242247.

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31

Tamassia, Roberto, and Ioannis G. Tollis. "Report on graph drawing '94." ACM SIGACT News 26, no. 1 (March 1995): 87–91. http://dx.doi.org/10.1145/203610.203615.

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32

Stone, Roger, Firat Batmaz, and Chris Hinde. "Drawing and Marking Graph Diagrams." Innovation in Teaching and Learning in Information and Computer Sciences 8, no. 2 (June 2009): 45–52. http://dx.doi.org/10.11120/ital.2009.08020045.

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33

Pisanski, Tomaž, and John Shawe-Taylor. "Characterizing Graph Drawing with Eigenvectors‡." Journal of Chemical Information and Computer Sciences 40, no. 3 (May 2000): 567–71. http://dx.doi.org/10.1021/ci9900938.

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34

Purchase, HELEN C. "Metrics for Graph Drawing Aesthetics." Journal of Visual Languages & Computing 13, no. 5 (October 2002): 501–16. http://dx.doi.org/10.1006/jvlc.2002.0232.

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35

Michailidis, George, and Jan de Leeuw. "Data Visualization through Graph Drawing." Computational Statistics 16, no. 3 (September 2001): 435–50. http://dx.doi.org/10.1007/s001800100077.

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36

Shiono, Yasunori, Toshihiro Yoshizumi, and Kensei Tsuchida. "Improvement of Fuzzy Graph Drawing Using Partition Tree." Journal of Advanced Computational Intelligence and Intelligent Informatics 26, no. 1 (January 20, 2022): 17–22. http://dx.doi.org/10.20965/jaciii.2022.p0017.

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Obtaining useful information from ambiguous information is a necessity in various fields. Ambiguous information can be handled quantitatively by using fuzzy theory, and representing it in an easy-to-understand manner is critical. One solution is to visualize an ambiguous relationship by using fuzzy graph representation, which has the essential characteristic of expressing variable relationships in between its nodes. We previously proposed an algorithm to draw intelligible and comprehensive fuzzy graphs. This study describes an improved drawing method for that graph drawing algorithm. As a result, highly related nodes were arranged closer to one another, and the display area was reduced. This method can be used as an effective means of expressing the results of ambiguous information analysis.

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37

FRATI, FABRIZIO. "ON MINIMUM AREA PLANAR UPWARD DRAWINGS OF DIRECTED TREES AND OTHER FAMILIES OF DIRECTED ACYCLIC GRAPHS." International Journal of Computational Geometry & Applications 18, no. 03 (June 2008): 251–71. http://dx.doi.org/10.1142/s021819590800260x.

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It has been shown that there exist planar digraphs that require exponential area in every upward straight-line planar drawing. On the other hand, upward poly-line planar drawings of planar graphs can be realized in Θ(n2) area. In this paper we consider families of DAGs that naturally arise in practice, like DAGs whose underlying graph is a tree (directed trees), is a bipartite graph (directed bipartite graphs), or is an outerplanar graph (directed outerplanar graphs). Concerning directed trees, we show that optimal Θ(n log n) area upward straight-line/poly-line planar drawings can be constructed. However, we prove that if the order of the neighbors of each node is assigned, then exponential area is required for straight-line upward drawings and quadratic area is required for poly-line upward drawings, results surprisingly and sharply contrasting with the area bounds for planar upward drawings of undirected trees. After having established tight bounds on the area requirements of planar upward drawings of several families of directed trees, we show how the results obtained for trees can be exploited to determine asymptotic optimal values for the area occupation of planar upward drawings of directed bipartite graphs and directed outerplanar graphs.

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38

TAMASSIA, ROBERTO, IOANNIS G. TOLLIS, and JEFFREY SCOTT VITTER. "A Parallel Algorithm for Planar Orthogonal Grid Drawings." Parallel Processing Letters 10, no. 01 (March 2000): 141–50. http://dx.doi.org/10.1142/s0129626400000147.

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In this paper we consider the problem of constructing planar orthogonal grid drawings (or more simply, layouts) of graphs, with the goal of minimizing the number of bends along the edges. We present optimal parallel algorithms that construct graph layouts with O(n) maximum edge length, O(n2) area, and at most 2n+4 bends (for biconnected graphs) and 2.4n+2 bends (for simply connected graphs). All three of these quality measures for the layouts are optimal in the worst case for biconnected graphs. The algorithm runs on a CREW PRAM in O( log n) time with n/ log n processors, thus achieving optimal time and processor utilization. Applications include VLSI layout, graph drawing, and wireless communication.

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39

Sokas, Algirdas. "Intelligent Agent Find its Way in the Drawing." Solid State Phenomena 165 (June 2010): 425–30. http://dx.doi.org/10.4028/www.scientific.net/ssp.165.425.

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This article analyzes intelligent agent in the changing drawing. The objective is to find the shortest way between two points in the flat space with prominent polygon fences. This is an idealized task that a robot (agent) has to solve seeking to find its way in the environment (drawing). The creation tasks of intelligent agent are solved with agent unified modeling language. Graphical system can analyze drawing, forming graph, calculate graph matrices, extract route and prepare programs form with information. It discerns objects-classes: agent, graph, route, which have some properties and methods. System test is executed with three drawings. Design system and example of intelligent agent in the drawing is presented. Intelligent agent systems are discussed and conclusions are made.

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40

ZHANG, Wei, Rui-bi ZENG, and Ming-xiao HU. "Weight-length consistent graph drawing algorithm for weighted undirected graphs." Journal of Computer Applications 32, no. 4 (April 9, 2013): 1116–18. http://dx.doi.org/10.3724/sp.j.1087.2012.01116.

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41

QUAN, Wu. "Fast Convergence Layout Algorithm for Drawing Graphs in Marching-Graph." Journal of Software 19, no. 8 (October 21, 2008): 1920–32. http://dx.doi.org/10.3724/sp.j.1001.2008.01920.

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42

SUDERMAN, MATTHEW. "PATHWIDTH AND LAYERED DRAWINGS OF TREES." International Journal of Computational Geometry & Applications 14, no. 03 (June 2004): 203–25. http://dx.doi.org/10.1142/s0218195904001433.

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An h-layer drawing of a graph G is a planar drawing of G in which each vertex is placed on one of h parallel lines and each edge is drawn as a straight line between its end-vertices. In such a drawing, we say that an edge is proper if its endpoints lie on adjacent layers, flat if they lie on the same layer and long otherwise. Thus, a proper h-layer drawing contains only proper edges, a short h-layer drawing contains no long edges, an upright h-layer drawing contains no flat edges, and an unconstrained h-layer drawing contains any type of edge. In this paper, we derive upper and lower bounds on the number of layers required by proper, short, upright, and unconstrained layered drawings of trees. We prove that these bounds are optimal with respect to the pathwidth of the tree being drawn. Finally, we give linear-time algorithms for obtaining layered drawings that match these upper bounds.

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43

Hua, Jie, Mao Lin Huang, Weidong Huang, and Chenglin Zhao. "Applying Graph Centrality Metrics in Visual Analytics of Scientific Standard Datasets." Symmetry 11, no. 1 (January 1, 2019): 30. http://dx.doi.org/10.3390/sym11010030.

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Graphs are often used to model data with a relational structure and graphs are usually visualised into node-link diagrams for a better understanding of the underlying data. Node-link diagrams represent not only data entries in a graph, but also the relations among the data entries. Further, many graph drawing algorithms and graph centrality metrics have been successfully applied in visual analytics of various graph datasets, yet little attention has been paid to analytics of scientific standard data. This study attempts to adopt graph drawing methods (force-directed algorithms) to visualise scientific standard data and provide information with importance ‘ranking’ based on graph centrality metrics such as Weighted Degree, PageRank, Eigenvector, Betweenness and Closeness factors. The outcomes show that our method can produce clear graph layouts of scientific standard for visual analytics, along with the importance ‘ranking’ factors (represent via node colour, size etc.). Our method may assist users with tracking various relationships while understanding scientific standards with fewer relation issues (missing/wrong connection etc.) through focusing on higher priority standards.

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Wang, Yiqiao, Juan Liu, Yongtang Shi, and Weifan Wang. "Star Chromatic Index of 1-Planar Graphs." Symmetry 14, no. 6 (June 8, 2022): 1177. http://dx.doi.org/10.3390/sym14061177.

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Many symmetric properties are well-explored in graph theory, especially in graph coloring, such as symmetric graphs defined by the automorphism groups, symmetric drawing of planar graphs, and symmetric functions which are used to count the number of specific colorings of a graph. This paper is devoted to studying the star edge coloring of 1-planar graphs. The star chromatic index χst′(G) of a graph G is defined as the smallest k for which the edges of G can be colored by using k colors so that no two adjacent edges get the same color and no bichromatic paths or cycles of length four are produced. A graph G is called 1-planar if it can be drawn in the plane such that each edge crosses at most one other edge. In this paper, we prove that every 1-planar graph G satisfies χst′(G)≤7.75Δ+166; and moreover χst′(G)≤⌊1.5Δ⌋+500 if G contains no 4-cycles, and χst′(G)≤2.75Δ+116 if G is 3-connected, or optimal, or NIC-planar.

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Raksha, M. R., P. Hithavarshini, Charles Dominic, and N. K. Sudev. "Injective coloring of complementary prism and generalized complementary prism graphs." Discrete Mathematics, Algorithms and Applications 12, no. 02 (February 28, 2020): 2050026. http://dx.doi.org/10.1142/s1793830920500263.

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The complementary prism [Formula: see text] of a graph [Formula: see text] is the graph obtained by drawing edges between the corresponding vertices of a graph [Formula: see text] and its complement [Formula: see text]. In this paper, we generalize the concept of complementary prisms of graphs and determine the injective chromatic number of generalized complementary prisms of graphs. We prove that for any simple graph [Formula: see text] of order [Formula: see text], [Formula: see text] and if [Formula: see text] is a graph with a universal vertex, then [Formula: see text].

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Nešetril, Jaroslav. "Art of Graph Drawing and Art." Journal of Graph Algorithms and Applications 6, no. 1 (2002): 131–47. http://dx.doi.org/10.7155/jgaa.00047.

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Gajer, Pawel, and Stephen G. Kobourov. "GRIP: Graph Drawing with Intelligent Placement." Journal of Graph Algorithms and Applications 6, no. 3 (2002): 203–24. http://dx.doi.org/10.7155/jgaa.00052.

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Harel, David, and Yehuda Koren. "Graph Drawing by High-Dimensional Embedding." Journal of Graph Algorithms and Applications 8, no. 2 (2004): 195–214. http://dx.doi.org/10.7155/jgaa.00089.

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Zheng, Jonathan X., Samraat Pawar, and Dan F. M. Goodman. "Graph Drawing by Stochastic Gradient Descent." IEEE Transactions on Visualization and Computer Graphics 25, no. 9 (September 1, 2019): 2738–48. http://dx.doi.org/10.1109/tvcg.2018.2859997.

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Sugiyama, Kozo. "A COGNITIVE APPROACH FOR GRAPH DRAWING." Cybernetics and Systems 18, no. 6 (January 1987): 447–88. http://dx.doi.org/10.1080/01969728708902150.

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