# Variants of the Segment Number of a Graph

The 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 establish lower and upper bounds on the new variants of the segment number, mostly for cubic graphs, depending on the connectivity of the given graph. We also 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.

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research
08/29/2023

### The Parametrized Complexity of the Segment Number

Given a straight-line drawing of a graph, a segment is a maximal set of ...
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### Drawing planar graphs with few segments on a polynomial grid

The visual complexity of a plane graph drawing is defined to be the numb...
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### Morphing Planar Graph Drawings Through 3D

In this paper, we investigate crossing-free 3D morphs between planar str...
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### Drawing Graphs on Few Circles and Few Spheres

Given a drawing of a graph, its visual complexity is defined as the numb...
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### Shortcut Hulls: Vertex-restricted Outer Simplifications of Polygons

Let P be a crossing-free polygon and 𝒞 a set of shortcuts, where each sh...
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### The maximum size of adjacency-crossing graphs

An adjacency-crossing graph is a graph that can be drawn such that every...
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### Polyline Drawings with Topological Constraints

Let G be a simple topological graph and let Γ be a polyline drawing of G...