Variants of the Segment Number of a Graph

TL;DR

This paper investigates different variants of the segment number in graph drawings, establishing complexity results and bounds, and comparing classical and new variants for planar and cubic graphs.

Contribution

It introduces and analyzes three variants of the segment number, proves their computational hardness, and provides bounds for cubic graphs based on connectivity.

Findings

Classical segment number can be asymptotically twice as large as new variants.

All variants are $orall ext{R}$-complete, implying NP-hardness.

Bounds are established for cubic graphs depending on connectivity.

Abstract

The \emph{segment number} of a planar graph is the smallest number of line segments whose union represents a crossing-free straight-line drawing of the given graph in the plane. The segment number is a measure for the visual complexity of a drawing; it has been studied extensively. In this paper, we study three variants of the segment number: for planar graphs, we consider crossing-free polyline drawings in 2D; for arbitrary graphs, we consider crossing-free straight-line drawings in 3D and straight-line drawings with crossings in 2D. We first construct an infinite family of planar graphs where the classical segment number is asymptotically twice as large as each of the new variants of the segment number. Then we establish the $\exists\mathbb{R}$-completeness (which implies the NP-hardness) of all variants. Finally, for cubic graphs, we prove lower and upper bounds on the new variants of the segment number, depending on the connectivity of the given graph.