Graph Treewidth and Geometric Thickness Parameters

Abstract

Consider a drawing of a graph G in the plane such that crossing edges are coloured differently. The minimum number of colours, taken over all drawings of G, is the classical graph parameter thickness. By restricting the edges to be straight, we obtain the geometric thickness. By additionally restricting the vertices to be in convex position, we obtain the book thickness. This paper studies the relationship between these parameters and treewidth.Our first main result states that for graphs of treewidth k, the maximum thickness and the maximum geometric thickness both equal ⌈k/2⌉. This says that the lower bound for thickness can be matched by an upper bound, even in the more restrictive geometric setting. Our second main result states that for graphs of treewidth k, the maximum book thickness equals k if k ≤ 2 and equals k + 1 if k ≥ 3. This refutes a conjecture of Ganley and Heath [Discrete Appl. Math. 109(3):215-221, 2001]. Analogous results are proved for outerthickness, arboricity, and star-arboricity.

Drawings of planar graphs with few slopes and segments

“…The next lemma says how to partition a 2-tree appropriately. It has subsequently been generalised for k-trees by Dujmović and Wood [11]. Otherwise G \ L is not a single edge, in which case, G \ L is a 2-tree.…”

“…This research was initiated at the International Workshop on Fixed Parameter Tractability in Geometry and Games, organised by Sue Whitesides; Bellairs Research Institute of McGill University, Barbados, February [7][8][9][10][11][12][13] 2004. Thanks to all of the participants for creating a stimulating working environment.…”

On graph thickness, geometric thickness, and separator theorems

“…Dujmović and Wood [26] discuss the relationship between geometric thickness and graph treewidth. In particular, they show that graphs with treewidth k have geometric thickness at most k/2 .…”

Partitions of complete geometric graphs into plane trees

“…Dillencourt et al [5] defined the geometric thickness of an (abstract) graph G to be the minimum k such that G has a representation as a geometric graph whose edges can be partitioned into k plane subgraphs; also see [3,7,8,10]. They proved that the geometric thickness of K n is between (n/5.646) + 0.342 and n/4 .…”