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arXiv · 1908.08273

Representing Graphs and Hypergraphs by Touching Polygons in 3D

William Evans·Paweł Rzążewski·Noushin Saeedi·Chan-Su Shin·Alexander Wolff
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Abstract

Contact representations of graphs have a long history. Most research has focused on problems in 2D, but 3D contact representations have also been investigated, mostly concerning fully-dimensional geometric objects such as spheres or cubes. In this paper we study contact representations with convex polygons in 3D. We show that every graph admits such a representation. Since our representations use super-polynomial coordinates, we also construct representations on grids of polynomial size for specific graph classes (bipartite, subcubic). For hypergraphs, we represent their duals, that is, each vertex is represented by a point and each edge by a polygon. We show that even regular and quite small hypergraphs do not admit such representations. On the other hand, the two smallest Steiner triple systems can be represented.

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William Evans, Paweł Rzążewski, Noushin Saeedi, Chan-Su Shin, Alexander Wolff. 2019-08-22. Representing Graphs and Hypergraphs by Touching Polygons in 3D. https://doi.org/10.1007/978-3-030-35802-0_2

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