Discrete & Computational Geometry

, Volume 37, Issue 4, pp 641–670

Graph Treewidth and Geometric Thickness Parameters

Article

DOI: 10.1007/s00454-007-1318-7

Cite this article as:
Dujmovic, V. & Wood, D. Discrete Comput Geom (2007) 37: 641. doi:10.1007/s00454-007-1318-7

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.

Copyright information

© Springer 2007

Authors and Affiliations

  1. 1.Department of Mathematics and Statistics, McGill UniversityMontreal, QuebecCanada H3A 2A7
  2. 2.Departament de Matematica Aplicada II, Universitat Politecnica de Catalunya08034 BarcelonaSpain

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