Relating Graph Thickness to Planar Layers and Bend Complexity

Abstract

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 R 2 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 Fáry'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 Fáry'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( √ 2 t · n 1−(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 3(k−1)n (4k−2) .

Simultaneous Embedding of Colored Graphs

“…Durocher and Mondal [10] showed that every thickness-k graph can be drawn on k planar layers with bend complexity at most O(…”

“…We then color all the division vertices with a color different than the input colors. Durocher and Mondal [10] showed that if P i admits an b-bend uphill drawing on a point set S, then G i admits a O(b)-bend polyline drawing on S. Hence it suffices to consider only the simultaneous embedding of the spinal paths. Here we give a concise proof for completeness.…”

“…Lemma 1 (Durocher and Mondal [10]). If a spinal path admits a b-bend uphill drawing on a point set S, then the corresponding graph admits an O(b)-bend planar polyline drawing on S.…”

“…This can be further improved by the following observations (Obs 1-2). These observations are made in [10], which generalize a standard technique of computing simultaneous geometric embedding of two paths [4] to drawing any number of paths with bends.…”

Edge partitions of optimal 2-plane and 3-plane graphs

“…For example, the arboricity of a graph G is the minimum number of forests needed to cover all edges of G, while G has thickness t if it is the union of t planar graphs. Durocher and Mondal [12] studied the interplay between the thickness t of a graph and the number of bends per edge in a drawing that can be partitioned into t planar sub-drawings.…”