Representing Graphs and Hypergraphs by Touching Polygons in 3D

TL;DR

This paper explores 3D contact representations of graphs and hypergraphs using convex polygons, demonstrating universality for graphs, limitations for hypergraphs, and polynomial-sized grid representations for specific classes.

Contribution

It proves that all graphs can be represented with convex polygons in 3D and constructs polynomial-sized grid representations for certain graph classes, while also analyzing hypergraph limitations.

Findings

All graphs admit convex polygon contact representations in 3D.

Hypergraph duals are not always representable with convex polygons in 3D.

Polynomial-sized grid representations exist for bipartite and subcubic graphs.

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.