用3D接触多边形表示图和超图
Representing Graphs and Hypergraphs by Touching Polygons in 3D
摘要：
图的接触表示有着悠久的历史。大多数研究集中在2D的问题，但3D接触表示也进行了研究，主要涉及全维几何对象，如球体或立方体。本文研究了三维凸多边形的接触表示。我们表明，每个图承认这样的表示。由于我们的表示使用超多项式坐标，我们也构造表示特定的图形类（二分，次立方）的多项式大小的网格。对于超图，我们表示它们的顶点，即每个顶点用一个点表示，每个边用一个多边形表示。我们表明，即使是经常和相当小的超图不承认这样的陈述。另一方面，可以表示两个最小的Steiner三元系。
英文摘要：
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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