The Community for Technology Leaders
RSS Icon
Subscribe
Issue No.06 - November/December (2010 vol.16)
pp: 1082-1089
Christian Heine , Institute for Computer Science, University of Leipzig, Germany
Marc Hellmuth , Institute for Computer Science, University of Leipzig, Germany
Peter F. Stadler , Institute for Computer Science, University of Leipzig, Germany
Gerik Scheuermann , Institute for Computer Science, University of Leipzig, Germany
ABSTRACT
Graphs are a versatile structure and abstraction for binary relationships between objects. To gain insight into such relationships, their corresponding graph can be visualized. In the past, many classes of graphs have been defined, e.g. trees, planar graphs, directed acyclic graphs, and visualization algorithms were proposed for these classes. Although many graphs may only be classified as "general" graphs, they can contain substructures that belong to a certain class. Archambault proposed the TopoLayout framework: rather than draw any arbitrary graph using one method, split the graph into components that are homogeneous with respect to one graph class and then draw each component with an algorithm best suited for this class. Graph products constitute a class that arises frequently in graph theory, but for which no visualization algorithm has been proposed until now. In this paper, we present an algorithm for drawing graph products and the aesthetic criterion graph product's drawings are subject to. We show that the popular High-Dimensional Embedder approach applied to cartesian products already respects this aestetic criterion, but has disadvantages. We also present how our method is integrated as a new component into the TopoLayout framework. Our implementation is used for further research of graph products in a biological context.
INDEX TERMS
Graph drawing, graph products, TopoLayout.
CITATION
Christian Heine, Marc Hellmuth, Peter F. Stadler, Gerik Scheuermann, "Visualization of Graph Products", IEEE Transactions on Visualization & Computer Graphics, vol.16, no. 6, pp. 1082-1089, November/December 2010, doi:10.1109/TVCG.2010.217
REFERENCES
[1] S. B. Akers and B. Krishnamurthy, A group-theoretic model for symmetric interconnection networks. IEEE Trans. Comput., 38 (4): 555–566, 1989.
[2] D. Archambault, T. Munzner, and D. Auber, Topolayout: Multilevel graph layout by topological features. IEEE Trans. Vis. Comput. Graph., 13 (2): 305–317, 2007.
[3] G. D. Battista, P. Eades, R. Tamassia, and I. Tollis, Graph Drawing. Prentice Hall, 1998.
[4] T. A. Davis, The University of Florida sparse matrix collection. Submitted to ACM Transactions on Mathematical Software.
[5] G. di Battista, P. Eades, R. Tamassia, and I. G. Tollis, Algorithms for drawing graphs: An annotated bibliography. Computational Geometry: Theory and Applications, 4 (5): 235–282, 1994.
[6] R. Diestel, Graph Theory (Graduate Texts in Mathematics). Springer, 2005.
[7] T. Dwyer, Scalable, versatile and simple constrained graph layout. Comput. Graph. Forum, 28 (3): 991–998, 2009.
[8] P. Eades, A heuristic for graph drawing. Congressus Numerantium, 42: 149–160, 1984.
[9] A. Frick, A. Ludwig, and H. Mehldau, A fast adaptive layout algorithm for undirected graphs. In R. Tamassia, and I. G. Tollis editors, Graph Drawing, volume 894 of LNCS, pages 388–403. Springer, 1994.
[10] Y. Frishman, and A. Tal, Multi-level graph layout on the GPU. IEEE Trans. Vis. Comput. Graph., 13 (6): 1310–1319, 2007.
[11] Y. Frishman and A. Tal, Online dynamic graph drawing. IEEE Trans. Vis. Comput. Graph., 14 (4): 727–740, 2008.
[12] T. M. J. Fruchterman and E. M. Reingold, Graph drawing by force-directed placement. Software - Practice and Experience, 21 (11): 1129–1164, 1991.
[13] P. Gajer and S. G. Kobourov, Grip: Graph drawing with intelligent placement. J. Graph Algorithms Appl., 6 (3): 203–224, 2002.
[14] E. R. Gansner, Y. Koren, and S. C. North, Graph drawing by stress majorization. In Pach [30], pages 239–250.
[15] S. Hachul and M. Junger, Drawing large graphs with a potential-field-based multilevel algorithm. In Pach [30], pages 285–295.
[16] S. Hachul and M. Jünger, An experimental comparison of fast algorithms for drawing general large graphs. In P. Healy, and N. S. Nikolov editors, Graph Drawing, volume 3843 of LNCS, pages 235–250. Springer, 2005.
[17] R. Hammack, and W. Imrich, On Cartesian skeletons of graphs. Ars Mathematica Contemporanea, 2 (2): 191–205, 2009.
[18] D. Harel and Y. Koren, A fast multi-scale method for drawing large graphs. J. Graph Algorithms Appl., 6 (3): 179–202, 2002.
[19] D. Harel and Y. Koren, Graph drawing by high-dimensional embedding. J. Graph Algorithms Appl., 8 (2): 195–214, 2004.
[20] M. Hellmuth, W. Imrich, W. Klöckl, and P. F. Stadler, Approximate graph products. European Journal of Combinatorics, 30: 119–1133, 2009.
[21] M. Hellmuth, W. Imrich, W. Klöckl, and P. F. Stadler, Local algorithms for the prime factorization of strong product graphs. Mathematics in Computer Science, 2 (4): 653–682, 2009.
[22] I. Herman, G. Melançon, and M. S. Marshall, Graph visualization and navigation in information visualization: A survey. IEEE Transactions on Visualization and Computer Graphics, 06 (1): 24–43, 2000.
[23] W. Imrich and S. Klavzar, Product Graphs. Wiley-Interscience, New-York, 2000.
[24] W. Imrich and I. Peterin, Recognizing Cartesian products in linear time. Discrete Math., 307: 472–482, 2007.
[25] T. Kamada and S. Kawai, An algorithm for drawing general undirected graphs. Inf. Process. Lett., 31 (1): 7–15, 1989.
[26] M. Kaufmann and D. Wagner editors. , Drawing graphs: methods and models. Springer-Verlag, London, UK, 2001.
[27] A. Kaveh and K. Koohestani, Graph products for configuration processing of space structures. Comput. Struct., 86 (11–12): 1219–1231, 2008.
[28] A. Kaveh and R. Mirzaie, Minimal cycle basis of graph products for the force method of frame analysis. Communications in Numerical Methods in Engineering, 24 (8): 653–669, 2008.
[29] A. Kaveh and H. Rahami, An efficient method for decomposition of regular structures using graph products. Intern. J.forNumer. Methods in Engineering, 61 (11): 1797–1808, 2004.
[30] J. Pach editor. Graph Drawing, 12th International Symposium, GD2004, New York, NY, USA, September 29 - October 2, 2004, Revised Selected Papers, volume 3383 of LNCS. Springer, 2004.
[31] H. C. Purchase, Which aesthetic has the greatest effect on human understanding? In G. D. Battista editor, Graph Drawing, volume 1353 of LNCS, pages 248–261. Springer, 1997.
[32] M. Schlosser, M. Sintek, S. Decker, and W. Nejdl, Hypercup - hyper-cubes, ontologies and efficient search on p2p networks. In LNCS, pages 112–124. Springer, 2002.
[33] B. M.R. Stadler and P. F. Stadler, The topology of evolutionary biology. In In Ciobanu, editor, Modeling in Molecular Biology, Natural Computing Series, pages 267–286. Springer Verlag, 2004.
[34] G. P. Wagner and P. F. Stadler, Quasi-independence, homology and the unity of type: A topological theory of characters. J. Theor. Biol, 220: 505–527, 2003.
[35] C. Walshaw, A multilevel algorithm for force-directed graph-drawing. J. Graph Algorithms Appl., 7 (3): 253–285, 2003.
23 ms
(Ver 2.0)

Marketing Automation Platform Marketing Automation Tool