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.Comment: 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, 200

Vertex partitions of chordal graphs

“…We generalise these additional properties in Section 3. The price paid is that each bag may now induce a (k − ℓ)-tree, thus matching the result of Dujmović et al [10,11] for ℓ = 1. Note that the proof of Dujmović et al [10,11] uses a different construction to the one given here.…”

“…Thus, it seems unavoidable that the maximum degree appears in an upper bound on the tree-partition-width. This fact, along with other applications, motivated Dujmović et al [10,11] to study the structure of the bags in a tree-partition. In this paper we continue this approach, and prove the following result (in Section 2).…”

“…Dujmović et al [10,11] proved that every k-tree has a tree-partition in which each bag induces a (k − 1)-tree. Thus Theorem 1 with ℓ = 1 improves this result.…”

“…Thus Theorem 1 with ℓ = 1 improves this result. That said, the tree-partition of Dujmović et al [10,11] has a number of additional properties that were important for the intended application. We generalise these additional properties in Section 3.…”

Untitled

“…A graph with a k-queue layout is called a k-queue graph. The queue-number of a graph G is the minimum integer k such that there is a k-queue layout of G. See [13,29,30,16,15,8,14,17] and the references therein for results on queue layouts. A d-monotone bipartite graph has queue-number at most d, since using the above notation, edges in a monotone matching do not cross in the vertex ordering (v 1 , .…”