Contact Representations of Graphs in 3D

TL;DR

This paper explores contact representations of graphs in 3D using boxes and L-shaped polyhedra, providing algorithms for representing certain classes of planar and non-planar graphs efficiently.

Contribution

It extends 2D primal-dual contact representations to 3D for 3-connected planar graphs and introduces algorithms for representing optimal 1-planar graphs with boxes and L-shaped polyhedra.

Findings

Existence of 3D box representations for 3-connected planar graphs and their duals.

Linear-time algorithms for constructing these representations.

Representation algorithms for optimal 1-planar graphs with boxes and L-shaped polyhedra.

Abstract

We study contact representations of graphs in which vertices are represented by axis-aligned polyhedra in 3D and edges are realized by non-zero area common boundaries between corresponding polyhedra. We show that for every 3-connected planar graph, there exists a simultaneous representation of the graph and its dual with 3D boxes. We give a linear-time algorithm for constructing such a representation. This result extends the existing primal-dual contact representations of planar graphs in 2D using circles and triangles. While contact graphs in 2D directly correspond to planar graphs, we next study representations of non-planar graphs in 3D. In particular we consider representations of optimal 1-planar graphs. A graph is 1-planar if there exists a drawing in the plane where each edge is crossed at most once, and an optimal n-vertex 1-planar graph has the maximum (4n - 8) number of edges. We describe a linear-time algorithm for representing optimal 1-planar graphs without separating 4-cycles with 3D boxes. However, not every optimal 1-planar graph admits a representation with boxes. Hence, we consider contact representations with the next simplest axis-aligned 3D object, L-shaped polyhedra. We provide a quadratic-time algorithm for representing optimal 1-planar graph with L-shaped polyhedra.