Three dimensional graph drawing with fixed vertices and one bend per edge

TL;DR

This paper proves that any graph with fixed vertex positions in 3D space can be drawn on a grid with only one bend per edge, improving the previous bound of three bends.

Contribution

It establishes a new theoretical bound showing that one bend per edge suffices for 3D grid drawings with fixed vertices, enhancing prior results.

Findings

Every graph with fixed vertices in D can be drawn with one bend per edge.

Improves the previous bound of three bends per edge.

Provides a theoretical foundation for more efficient 3D graph drawings.

Abstract

We prove that for every graph $G$, given fixed locations for the vertices of $G$ in $\mathbb{Z}^3$, there is a three-dimensional grid-drawing of $G$ with one bend per edge. The best previous bound was three bends per edge.