Layout of Graphs with Bounded Tree-Width

Abstract

A \emph{queue layout} of a graph consists of a total order of the vertices, and a partition of the edges into \emph{queues}, such that no two edges in the same queue are nested. The minimum number of queues in a queue layout of a graph is its \emph{queue-number}. A \emph{three-dimensional (straight-line grid) drawing} of a graph represents the vertices by points in ${\mathbb{Z}}^3$ and the edges by non-crossing line-segments. This paper contributes three main results: (1) It is proved that the minimum volume of a certain type of three-dimensional drawing of a graph $G$ is closely related to the queue-number of $G$. In particular, if $G$ is an $n$-vertex member of a proper minor-closed family of graphs (such as a planar graph), then $G$ has a $O(1)\times O(1)\times O(n)$ drawing if and only if $G$ has O(1) queue-number. (2) It is proved that queue-number is bounded by tree-width, thus resolving an open problem due to Ganley and Heath (2001), and disproving a conjecture of Pemmaraju (1992). This result provides renewed hope for the positive resolution of a number of open problems in the theory of queue layouts. (3) It is proved that graphs of bounded tree-width have three-dimensional drawings with O(n) volume. This is the most general family of graphs known to admit three-dimensional drawings with O(n) volume. The proofs depend upon our results regarding \emph{track layouts} and \emph{tree-partitions} of graphs, which may be of independent interest.

This is a revised version of a journal paper submitted in October 2002. This paper incorporates the following conference papers: (1) Dujmovic', Morin & Wood. Path-width and three-dimensional straight-line grid drawings of graphs (GD'02), LNCS 2528:42-53, Springer, 2002. (2) Wood. Queue layouts, tree-width, and three-dimensional graph drawing (FSTTCS'02), LNCS 2556:348--359, Springer, 2002. (3) Dujmovic' & Wood. Tree-partitions of $k$-trees with applications in graph layout (WG '03), LNCS 2880:205-217, 2003

Cited by in corpus (25)

• Characterisations and Examples of Graph Classes with Bounded Expansion
• Planar graphs have bounded queue-number
• Layered Separators in Minor-Closed Graph Classes with Applications
• On Tree-Partition-Width
• Graph Drawings with Few Slopes
• Improved bounds for centered colorings
• Track Layouts, Layered Path Decompositions, and Leveled Planarity
• Planar graphs have bounded nonrepetitive chromatic number
• Stack-number is not bounded by queue-number
• Minor-closed graph classes with bounded layered pathwidth
• Upward Three-Dimensional Grid Drawings of Graphs
• Track Layouts of Graphs
• Book Embeddings of Graph Products
• Queue Layouts of Graphs with Bounded Degree and Bounded Genus
• Bad News for Chordal Partitions
• Planar Graphs of Bounded Degree have Constant Queue Number
• Thickness and Antithickness of Graphs
• Notes on Nonrepetitive Graph Colouring
• Layouts for Plane Graphs on Constant Number of Tracks
• Parameters Tied to Treewidth
• Local and Union Boxicity
• Planar Decompositions and the Crossing Number of Graphs with an Excluded Minor
• Drawing a Graph in a Hypercube
• Crossings in Grid Drawings
• Queue layouts and nonrepetitive colouring of planar graphs and powers of trees