On Representing Graphs by Touching Cuboids

Abstract

Abstract.We consider contact representations of graphs where vertices are represented by cuboids, i.e. interior-disjoint axis-aligned boxes in 3D space. Edges are represented by a proper contact between the cuboids representing their endvertices. Two cuboids make a proper contact if they intersect and their intersection is a non-zero area rectangle contained in the boundary of both. We study representations where all cuboids are unit cubes, where they are cubes of different sizes, and where they are axis-aligned 3D boxes. We prove that it is NP-complete to decide whether a graph admits a proper contact representation by unit cubes. We also describe algorithms that compute proper contact representations of varying size cubes for relevant graph families. Finally, we give two new simple proofs of a theorem by Thomassen stating that all planar graphs have a proper contact representation by touching cuboids.

On Contact Graphs with Cubes and Proportional Boxes

“…Note that by moving the cubes in the representation by an arbitrarily small amount, we can make sure that for each triangle xyz in P * , the three cubes for x, y and z form a corner surrounded by three mutually touching walls at right angles to each other. Now observe that each of the three graphs G a , G b and G c is a planar 3-tree; thus using the algorithm of either [5] or [10], we can place the internal vertices of these three graphs in their corresponding corners, thereby completing the representation.…”

“…Thomassen [17] shows that any planar graph has a proper contact representation with touching boxes, while Felsner and Francis [10] find a (not necessarily proper) contact representation of any planar graph with touching cubes. Recently, Bremner et al [5] asked whether any planar graph can be represented by proper contacts of cubes. They answered the question positively for the case of partial planar 3-trees and some planar grids, but the problem remains open for general planar graphs.…”

“…We also show that several classes of planar graphs admit proportional box-contact representations. Specifically, we show how to compute a proportional box-contact representation for plane 3-trees, while a cube-contact representation for the same graph class follows from [5]. We also show how to compute a proportional box-contact representation and a cube-contact representation for nested maximal outerplanar graphs, which are defined as follows.…”

3D proportional contact representations of graphs

“…On the other hand, there is much less known about contact representations in the 3D space. Contact graphs using 3D objects have been studied for spheres [4], [18], cylinders [3], tetrahedra [2] and axis-aligned cuboids [6], [15], [22]. Thomassen shows that any planar graph has a contact representation by cuboids [22].…”

“…For axis-aligned polyhedra, the notion of polyhedral complexity can be measured by the number of cuboids that make a polyhedra, or by the number of corners, edges and faces in the polyhedra. Previous results on axis-aligned contact representations in 3D only consider axisaligned cuboids to represent vertices [6], [15], [22]. However, there are graphs, even as simple as K 5 , for which cuboids are not sufficient.…”