Graph Treewidth and Geometric Thickness Parameters

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 further 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 $\lceil{k/2}\rceil$. 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 $k \leq 2$ and equals $k+1$ if $k \geq 3$. 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.Comment: A preliminary version of this paper appeared in the "Proceedings of the 13th International Symposium on Graph Drawing" (GD '05), Lecture Notes in Computer Science 3843:129-140, Springer, 2006. The full version was published in Discrete & Computational Geometry 37(4):641-670, 2007. That version contained a false conjecture, which is corrected on page 26 of this versio

Musubime to karamime no gurafu hyogen

制度:新 ; 報告番号:乙2277号 ; 学位の種類:博士(理学) ; 授与年月日:2010/6/22 ; 早大学位記番号:新540

On the queue number of planar graphs

We prove that planar graphs have O(log2 n) queue number, thus improving upon the previous O(Formula Presented) upper bound. Consequently, planar graphs admit three-dimensional straight-line crossing-free grid drawings in O(n log8 n) volume, thus improving upon the previous O(n3/2) upper bound. © 2013 Society for Industrial and Applied Mathematics