 License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2016.10
URN: urn:nbn:de:0030-drops-62767
URL: https://drops.dagstuhl.de/opus/volltexte/2016/6276/
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### Relating Graph Thickness to Planar Layers and Bend Complexity

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### 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).

### BibTeX - Entry

```@InProceedings{durocher_et_al:LIPIcs:2016:6276,
author =	{Stephane Durocher and Debajyoti Mondal},
title =	{{Relating Graph Thickness to Planar Layers and Bend Complexity}},
booktitle =	{43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)},
pages =	{10:1--10:13},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-013-2},
ISSN =	{1868-8969},
year =	{2016},
volume =	{55},
editor =	{Ioannis Chatzigiannakis and Michael Mitzenmacher and Yuval Rabani and Davide Sangiorgi},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address =	{Dagstuhl, Germany},
URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/6276},
URN =		{urn:nbn:de:0030-drops-62767},
doi =		{10.4230/LIPIcs.ICALP.2016.10},
annote =	{Keywords: Graph Drawing, Thickness, Geometric Thickness, Layers; Bends}
}
```

 Keywords: Graph Drawing, Thickness, Geometric Thickness, Layers; Bends Collection: 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016) Issue Date: 2016 Date of publication: 23.08.2016

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