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 thicknesst graphs can be drawn on t layers with 2.25n + O(1) bends per edge. We then develop another technique to draw thicknesst 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*(k1)*n/(4k2).
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 = {10:110:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770132},
ISSN = {18688969},
year = {2016},
volume = {55},
editor = {Ioannis Chatzigiannakis and Michael Mitzenmacher and Yuval Rabani and Davide Sangiorgi},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2016/6276},
URN = {urn:nbn:de:0030drops62767},
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 