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# Relating Graph Thickness to Planar Layers and Bend Complexity

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LIPIcs.ICALP.2016.10.pdf
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## Cite As

Stephane Durocher and Debajyoti Mondal. Relating Graph Thickness to Planar Layers and Bend Complexity. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 10:1-10:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
https://doi.org/10.4230/LIPIcs.ICALP.2016.10

## 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 Gamma 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 Gamma is the maximum number of bends per edge in Gamma, and the layer complexity of Gamma is the minimum integer r such that the set of polygonal chains in Gamma 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)). 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 reduced bend complexity for small values of t, e.g., for t in {3, 4}, the bend complexity decreases to O(sqrt(n)). 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).
##### Keywords
• Graph Drawing
• Thickness
• Geometric Thickness
• Layers; Bends

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