3D Visibility Representations of 1-planar Graphs

TL;DR

This paper proves that all 1-planar graphs can be represented in 3D using z-parallel visibilities with rectangles, linking 3D visibility to bar 1-visibility in a novel way.

Contribution

It introduces a new 3D visibility representation for 1-planar graphs that connects to bar 1-visibility through a intersecting plane.

Findings

Every 1-planar graph admits a z-parallel visibility representation.

The representation ensures a plane intersects all rectangles, creating a bar 1-visibility representation.

The construction provides a new geometric perspective on 1-planar graphs.

Abstract

We prove that every 1-planar graph G has a z-parallel visibility representation, i.e., a 3D visibility representation in which the vertices are isothetic disjoint rectangles parallel to the xy-plane, and the edges are unobstructed z-parallel visibilities between pairs of rectangles. In addition, the constructed representation is such that there is a plane that intersects all the rectangles, and this intersection defines a bar 1-visibility representation of G.