Representing Graphs and Hypergraphs by Touching Polygons in 3D
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.
Cite
@article{arxiv.1908.08273,
title = {Representing Graphs and Hypergraphs by Touching Polygons in 3D},
author = {William Evans and Paweł Rzążewski and Noushin Saeedi and Chan-Su Shin and Alexander Wolff},
journal= {arXiv preprint arXiv:1908.08273},
year = {2020}
}
Comments
Appeared in the Proceedings of the 27th International Symposium on Graph Drawing and Network Visualization (GD 2019)