Abstract
Graph and cartographic visualization have the common objective to provide intuitive understanding of some underlying data. We consider a problem that combines aspects of both by studying the problem of fitting planar graphs on planar maps. After providing an NP-hardness result for the general decision problem, we identify sufficient conditions so that a fit is possible on a map with rectangular regions. We generalize our techniques to non-convex rectilinear polygons, where we also address the problem of efficient distribution of the vertices inside the map regions.
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References
Angelini, P., Frati, F., Kaufmann, M.: Straight-line rectangular drawings of clustered graphs. In: Symposium on Algorithms & Data Structures (WADS), pp. 25–36 (2009)
Battista, G.D., Drovandi, G., Frati, F.: How to draw a clustered tree. Journal of Discrete Algorithms 7(4), 479–499 (2009)
Battista, G.D., Frati, F.: Efficient c-planarity testing for embedded flat clustered graphs with small faces. Journal of Graph Algorithms and Applications 13(3), 349–378 (2009)
Bern, M.W., Gilbert, J.R.: Drawing the planar dual. Information Processing Letters 43(1), 7–13 (1992)
Bruls, M., Huizing, K., van Wijk, J.J.: Squarified treemaps. In: Joint Eurographics and IEEE TCVG Symposium on Visualization, pp. 33–42 (2000)
Buchsbaum, A., Gansner, E., Procopiuc, C., Venkatasubramanian, S.: Rectangular layouts and contact graphs. ACM Transactions on Algorithms 4(1) (2008)
Chambers, E., Eppstein, D., Goodrich, M., Löffler, M.: Drawing graphs in the plane with a prescribed outer face and polynomial area. Journal of Graph Algorithms and Applications 16(2), 243–259 (2012)
Coffey, J.W., Hoffman, R.R., Cañas, A.J.: Concept map-based knowledge modeling: perspectives from information and knowledge visualization. Information Visualization 5(3), 192–201 (2006)
Duncan, C., Gansner, E., Hu, Y., Kaufmann, M., Kobourov, S.G.: Optimal polygonal representation of planar graphs. Algorithmica 63(3), 672–691 (2012)
Eades, P., Feng, Q.-W., Lin, X., Nagamochi, H.: Straight-line drawing algorithms for hierarchical graphs and clustered graphs. Algorithmica 44(1), 1–32 (2006)
Felsner, S.: Rectangle and square representations of planar graphs. In: Pach, J. (ed.) Thirty Essays on Geometric Graph Theory. Springer (2012)
Feng, Q.-W., Cohen, R.F., Eades, P.: How to draw a planar clustered graph. In: Li, M., Du, D.-Z. (eds.) COCOON 1995. LNCS, vol. 959, pp. 21–30. Springer, Heidelberg (1995)
Feng, Q.-W., Cohen, R.F., Eades, P.: Planarity for clustered graphs. In: European Symposium on Algorithms (ESA), pp. 213–226 (1995)
Fusy, É.: Transversal structures on triangulations: A combinatorial study and straight-line drawings. Discrete Mathematics 309(7), 1870–1894 (2009)
Harel, D.: On visual formalisms. Communications of the ACM 31(5), 514–530 (1988)
He, X.: On floor-plan of plane graphs. SIAM Journal of Computing 28(6), 2150–2167 (1999)
Hu, Y., Gansner, E.R., Kobourov, S.G.: Visualizing graphs and clusters as maps. IEEE Computer Graphics and Applications 30(6), 54–66 (2010)
Kamada, T.: Visualizing Abstract Objects and Relations. World Scientific Series in Computer Science, vol. 5 (1989)
Kerber, M.: Embedding the dual complex of hyper-rectangular partitions. Journal of Computational Geometry 4(1), 13–37 (2013)
Knuth, D.E., Raghunathan, A.: The problem of compatible representatives. SIAM Journal on Discrete Mathematics 5(3), 422–427 (1992)
Koźmiński, K., Kinnen, E.: Rectangular duals of planar graphs. Networks 15, 145–157 (1985)
Liao, C.-C., Lu, H.-I., Yen, H.-C.: Compact floor-planning via orderly spanning trees. Journal of Algorithms 48, 441–451 (2003)
Lichtenstein, D.: Planar formulae and their uses. SIAM Journal on Computing 11(2), 329–343 (1982)
Shneiderman, B.: Tree visualization with tree-maps: 2-D space-filling approach. ACM Transactions on Graphics 11(1), 92–99 (1992)
Ungar, P.: On diagrams representing graphs. Journal of London Mathematical Sociery 28, 336–342 (1953)
Yeap, K.-H., Sarrafzadeh, M.: Floor-planning by graph dualization: 2-concave rectilinear modules. SIAM Journal on Computing 22, 500–526 (1993)
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Alam, M.J., Kaufmann, M., Kobourov, S.G., Mchedlidze, T. (2014). Fitting Planar Graphs on Planar Maps. In: Geffert, V., Preneel, B., Rovan, B., Štuller, J., Tjoa, A.M. (eds) SOFSEM 2014: Theory and Practice of Computer Science. SOFSEM 2014. Lecture Notes in Computer Science, vol 8327. Springer, Cham. https://doi.org/10.1007/978-3-319-04298-5_6
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DOI: https://doi.org/10.1007/978-3-319-04298-5_6
Publisher Name: Springer, Cham
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