GeometricThickness2007

https://www.doi.org/10.1007/S00454-007-1318-7

@article{GeometricThickness2007,
author = {Vida Dujmovic and David R. Wood},
doi = {10.1007/S00454-007-1318-7},
issn = {1432-0444},
journaltitle = {Discret. Comput. Geom.},
number = {4},
pages = {641--670},
title = {Graph Treewidth and Geometric Thickness Parameters},
volume = {37},
year = {2007},
}

• page 3 : thickness – The thickness of a graph $G$, …, is the minimum number of planar subgraphs that partition (ed: edges of) $G$.
• page 3 : outerthickness – The outerthickness of a graph $G$, …, is the minimum number of outerplanar subgraphs that partition (ed: edges of) $G$.
• page 4 : treewidth $k$ upper bounds thickness by $\mathcal O(k)$ – Proposition 1. The maximum thickness of a graph in $\mathcal T_k$ (ed: $k$-tree) is $\lceil k/2 \rceil$; …
• page 5 : treewidth $k$ upper bounds arboricity by $\mathcal O(k)$ – Proposition 2. The maximum arboricity of a graph in $\mathcal T_k$ (ed: $k$-tree) is $k$; …
• page 5 : treewidth $k$ upper bounds outerthickness by $\mathcal O(k)$ – Proposition 3. The maximum outerthickness of a graph in $\mathcal T_k$ (ed: $k$-tree) is $k$; …
• page 6 : treewidth $k$ upper bounds star-arboricity by $\mathcal O(k)$ – Proposition 4. The maximum star-arboricity of a graph in $\mathcal T_k$ (ed: $k$-tree) is $k+1$; …
• page 7 : book thickness – A geometric drawing in which the vertices are in convex position is called a book embedding. The book thickness of a graph $G$, …, is the minimum $k \in \mathbb N$ such that there is book embedding of $G$ with thickness $k$.
• page 8 : treewidth $k$ upper bounds book thickness by $\mathcal O(k)$ – The maximum book thickness … of a graph $\mathcal T_k$ (ed: $k$-tree) satisfy … $=k$ for $k \le 2$, $=k+1$ for $k \ge 3$.