Drawing Graphs on Few Lines and Few Planes

TL;DR

This paper explores how to draw graphs in 2D and 3D with edges or vertices covered by few lines or planes, establishing bounds and relations to graph properties, especially for planar graphs.

Contribution

It provides the first comprehensive analysis of covering graph drawings with few lines or planes, including bounds, exact values, and relations to graph parameters.

Findings

Bounds for lines and planes needed for graph drawings

Exact values for certain graph classes

Planar graphs can be drawn with fewer lines in 3D than in 2D

Abstract

We investigate the problem of drawing graphs in 2D and 3D such that their edges (or only their vertices) can be covered by few lines or planes. We insist on straight-line edges and crossing-free drawings. This problem has many connections to other challenging graph-drawing problems such as small-area or small-volume drawings, layered or track drawings, and drawing graphs with low visual complexity. While some facts about our problem are implicit in previous work, this is the first treatment of the problem in its full generality. Our contribution is as follows. We show lower and upper bounds for the numbers of lines and planes needed for covering drawings of graphs in certain graph classes. In some cases our bounds are asymptotically tight; in some cases we are able to determine exact values. We relate our parameters to standard combinatorial characteristics of graphs (such as the chromatic number, treewidth, maximum degree, or arboricity) and to parameters that have been studied in graph drawing (such as the track number or the number of segments appearing in a drawing). We pay special attention to planar graphs. For example, we show that there are planar graphs that can be drawn in 3-space on a lot fewer lines than in the plane.