The thickness of a graph G=(V,E) with n vertices is the minimum number of
planar subgraphs of G whose union is G. A polyline drawing of G in
R2 is a drawing Γ of G, where each vertex is mapped to a
point and each edge is mapped to a polygonal chain. Bend and layer complexities
are two important aesthetics of such a drawing. The bend complexity of Γ
is the maximum number of bends per edge in Γ, and the layer complexity
of Γ is the minimum integer r such that the set of polygonal chains in
Γ can be partitioned into r disjoint sets, where each set corresponds
to a planar polyline drawing. Let G be a graph of thickness t. By
F\'{a}ry's theorem, if t=1, then G can be drawn on a single layer with bend
complexity 0. A few extensions to higher thickness are known, e.g., if t=2
(resp., t>2), then G can be drawn on t layers with bend complexity 2
(resp., 3n+O(1)). However, allowing a higher number of layers may reduce the
bend complexity, e.g., complete graphs require Θ(n) layers to be drawn
using 0 bends per edge.
In this paper we present an elegant extension of F\'{a}ry's theorem to draw
graphs of thickness t>2. We first prove that thickness-t graphs can be
drawn on t layers with 2.25n+O(1) bends per edge. We then develop another
technique to draw thickness-t graphs on t layers with bend complexity,
i.e., O(2t⋅n1−(1/β)), where β=2⌈(t−2)/2⌉. Previously, the bend complexity was not known to be sublinear for
t>2. Finally, we show that graphs with linear arboricity k can be drawn on
k layers with bend complexity (4k−2)3(k−1)n.Comment: A preliminary version appeared at the 43rd International Colloquium
on Automata, Languages and Programming (ICALP 2016