3D proportional contact representations of graphs

Abstract

Abstract-In 3D contact representations, the vertices of a graph are represented by 3D polyhedra and the edges are realized by non-zero-area common boundaries between corresponding polyhedra. While contact representations with cuboids have been studied in the literature, we consider a novel generalization of the problem in which vertices are represented by axis-aligned polyhedra that are union of two cuboids. In particular, we study the weighted (proportional) version of the problem, where the volumes of the polyhedra and the areas of the common boundaries realize prespecified vertex and edge weights. For some classes of graphs (e.g., outerplanar, planar bipartite, planar, complete), we provide algorithms to construct such representations for arbitrary given weights. We also show that not all graphs can be represented in 3D with axis-aligned polyhedra of constant complexity.

On Contact Graphs with Cubes and Proportional Boxes

“…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. The weighted variant of the problem in 3D is much less studied, although recently Alam et al [1] have presented algorithms for proportional representation of several classes of graphs (e.g., outerplanar, planar bipartite, planar, complete), using 3D L-shapes. Our Contribution: Here we expand the class of planar graph representable by proper contact of cubes.…”