Abstract
As analogs to grid intersection graphs and rectangle intersection graphs in the plane, we consider grid intersection graphs, grid contact graphs and box intersection graphs on the other two euclidean surfaces — the annulus and the torus. Our first results concern the inclusions among these classes, and the main result is negative — there are bipartite box intersection graphs on annulus (torus), which are not grid intersection graphs on the particular surfaces (in contrast to the planar case, where the two classes are equal, cf. Bellantoni, Hartman, Przytycka, Whitesides: Grid intersection graphs and boxicity, Discrete Math. 114 (1993), 41–49). We also consider the question of computational complexity of recognizing these classes. Among other results, we show that recognition of grid intersection graphs on annulus and torus are both polynomial time solvable, provided orderings of both vertical and horizontal segments are specified.
This research was started when the first author was visiting Odense University in May 1994. The paper was written when the first author visited University of Oregon in Eugene, Oregon in the academic year 1994/5. The first author also acknowledges partial support from Czech Research Grants GAUK 361 and GAČR 2167.
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© 1996 Springer-Verlag Berlin Heidelberg
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Kratochvíl, J., Przytycka, T. (1996). Grid intersection and box intersection graphs on surfaces. In: Brandenburg, F.J. (eds) Graph Drawing. GD 1995. Lecture Notes in Computer Science, vol 1027. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0021820
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DOI: https://doi.org/10.1007/BFb0021820
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