On Contact Graphs with Cubes and Proportional Boxes

TL;DR

This paper explores methods to represent planar graphs using contact graphs of cubes and proportional boxes, providing algorithms for specific graph classes to enhance visualization and spatial understanding.

Contribution

It introduces algorithms for constructing cube-contact and proportional box-contact representations for various classes of planar graphs, advancing graph visualization techniques.

Findings

Algorithms successfully construct contact representations for certain planar graph classes.

Proportional box-contact representations accommodate prescribed volumes, enabling customizable visualizations.

The methods improve understanding of spatial relationships in planar graphs.

Abstract

We study two variants of the problem of contact representation of planar graphs with axis-aligned boxes. In a cube-contact representation we realize each vertex with a cube, while in a proportional box-contact representation each vertex is an axis-aligned box with a prespecified volume. We present algorithms for constructing cube-contact representation and proportional box-contact representation for several classes of planar graphs.