Abstract
A three-dimensional grid drawing of a graph G is a placement of the vertices at distinct integer points so that the straight-line segments representing the edges of G are pairwise non-crossing. It is shown that for any fixed r ≥ 2, every r-colorable graph of n vertices has a three-dimensional grid drawing that fits into a box of volume O(n 2). The order of magnitude of this bound cannot be improved.
Supported by NSF grant CCR-94-24398 and PSC-CUNY Research Award 667339
Supported by the Deutsche Forschungsgemeinschaft
Supported by OTKA-T-020914, OTKA-F-22234 and the Alfred Sloan Foundation
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References
G. Di Battista, P. Eades, R. Tamassia, and I.G. Tollis, Algorithms for drawing graphs: an annotated bibliography, Computational Geometry: Theory and Applications 4 (1994), 235–282.
R.F. Cohen, P. Eades, T. Lin, and F. Ruskey, Three-dimensional graph drawing, in: Graph Drawing (GD '94, R. Tamassia and I.G. Tollis, eds.), Lecture Notes in Computer Science 1027, Springer-Verlag, Berlin, 1995, 1–11.
T. Calamoneri and A. Sterbini, Drawing 2-, 3-, and 4-colorable graphs in O(n 2) volume, in: Graph Drawing (GD '96, S. North, ed.), Lecture Notes in Computer Science 1190, Springer-Verlag, Berlin, 1997, 53–62.
H. de Fraysseix, J. Pach, and R. Pollack, How to draw a planar graph on a grid, Combinatorica 10 (1990), 41–51.
G. H. Hardy, E. M. Wright, An Introduction to the Theory of Numbers, University Press, Oxford, 1954.
J. Mackinley, G. Robertson, and S. Card, Cone trees: Animated 3d visualizations of hierarchical information, in: Proceedings of SIGCHI Conference on Human Factors in Computing, 1991, 189–194.
W. Schnyder, Embedding planar graphs on the grid, in: Proceedings of the 1st Annual ACM-SIAM Symposium on Discrete Algorithms, 1990, 138–148.
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© 1997 Springer-Verlag Berlin Heidelberg
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Pach, J., Thiele, T., Tóth, G. (1997). Three-dimensional grid drawings of graphs. In: DiBattista, G. (eds) Graph Drawing. GD 1997. Lecture Notes in Computer Science, vol 1353. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-63938-1_49
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DOI: https://doi.org/10.1007/3-540-63938-1_49
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