Relating Graph Thickness to Planar Layers and Bend Complexity

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 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 · n1−(1/β)), where β = 2d(t−2)/2e. 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−1n 4·3k−2+1 .