Sublogarithmic distributed MIS algorithm for sparse graphs using nash-williams decomposition
Proceedings of the twenty-seventh ACM symposium on Principles of distributed computing
A self-stabilizing algorithm for cut problems in synchronous networks
Theoretical Computer Science
On graph thickness, geometric thickness, and separator theorems
Computational Geometry: Theory and Applications
Compact navigation and distance oracles for graphs with small treewidth
ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
Graph treewidth and geometric thickness parameters
GD'05 Proceedings of the 13th international conference on Graph Drawing
Drawing cubic graphs with the four basic slopes
GD'11 Proceedings of the 19th international conference on Graph Drawing
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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.