The Topology of Bendless Three-Dimensional Orthogonal Graph Drawing

TL;DR

This paper explores a special class of 3-regular graphs called xyz graphs, establishing their topological properties, characterizing planar xyz graphs, and proving the NP-completeness of recognizing such graphs.

Contribution

It introduces a topological characterization of xyz graphs, links bipartiteness to surface orientability, and provides an algorithm for recognition testing.

Findings

Planar xyz graphs are exactly bipartite, cubic, and three-connected.

Recognizing xyz graphs is NP-complete.

An algorithm with O(n 2^{n/2}) complexity for testing xyz graphs.

Abstract

We consider embeddings of 3-regular graphs into 3-dimensional Cartesian coordinates, in such a way that two vertices are adjacent if and only if two of their three coordinates are equal (that is, if they lie on an axis-parallel line) and such that no three points lie on the same axis-parallel line; we call a graph with such an embedding an xyz graph}. We describe a correspondence between xyz graphs and face-colored embeddings of the graph onto two-dimensional manifolds, and we relate bipartiteness of the xyz graph to orientability of the underlying topological surface. Using this correspondence, we show that planar graphs are xyz graphs if and only if they are bipartite, cubic, and three-connected, and that it is NP-complete to determine whether an arbitrary graph is an xyz graph. We also describe an algorithm with running time O(n 2^{n/2}) for testing whether a given graph is an xyz graph.