  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.

Computational Geometry;Discrete Mathematics