We consider routing problems in ad hoc wireless networks modeled as unit graphs in which nodes are points in the plane and two nodes can communicate if the distance between them is less than some fixed unit. We describe the first distributed algorithms for routing that do not require duplication of packets or memory at the nodes and yet guaranty that a packet is delivered to its destination. These algorithms can be extended to yield algorithms for broadcasting and geocasting that do not require packet duplication. A byproduct of our results is a simple distributed protocol for extracting a planar subgraph of a unit graph. We also present simulation results on the performance of our algorithms.

We show that planar graphs have bounded queue-number, thus proving a conjecture of Heath et al. [66] from 1992. The key to the proof is a new structural tool called layered partitions , and the result that every planar graph has a vertex-partition and a layering, such that each part has a bounded number of vertices in each layer, and the quotient graph has bounded treewidth. This result generalises for graphs of bounded Euler genus. Moreover, we prove that every graph in a minor-closed class has such a layered partition if and only if the class excludes some apex graph. Building on this work and using the graph minor structure theorem, we prove that every proper minor-closed class of graphs has bounded queue-number. Layered partitions have strong connections to other topics, including the following two examples. First, they can be interpreted in terms of strong products. We show that every planar graph is a subgraph of the strong product of a path with some graph of bounded treewidth. Similar statements hold for all proper minor-closed classes. Second, we give a simple proof of the result by DeVos et al. [31] that graphs in a proper minor-closed class have low treewidth colourings.

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

We consider online routing algorithms for routing between the vertices of embedded planar straight line graphs. Our results include (1) two deterministic memoryless routing algorithms, one that works for all Delaunay triangulations and the other that works for all regular triangulations; (2) a randomized memoryless algorithm that works for all triangulations; (3) an O(1) memory algorithm that works for all convex subdivisions; (4) an O(1) memory algorithm that approximates the shortest path in Delaunay triangulations; and (5) theoretical and experimental results on the competitiveness of these algorithms.

Graph separators are a ubiquitous tool in graph theory and computer science. However, in some applications, their usefulness is limited by the fact that the separator can be as large as Ω( √ n) in graphs with n vertices. This is the case for planar graphs, and more generally, for proper minor-closed classes. We study a special type of graph separator, called a layered separator, which may have linear size in n, but has bounded size with respect to a different measure, called the width. We prove, for example, that planar graphs and graphs of bounded Euler genus admit layered separators of bounded width. More generally, we characterise the minor-closed classes that admit layered separators of bounded width as those that exclude a fixed apex graph as a minor.We use layered separators to prove O(log n) bounds for a number of problems where O( √ n) was a long-standing previous best bound. This includes the nonrepetitive chromatic number and queue-number of graphs with bounded Euler genus. We extend these results with a O(log n) bound on the nonrepetitive chromatic number of graphs excluding a fixed topological minor, and a log O(1) n bound on the queue-number of graphs excluding a fixed minor. Only for planar graphs were log O(1) n bounds previously known. Our results imply that every n-vertex graph excluding a fixed minor has a 3-dimensional grid drawing with n log O(1) n volume, whereas the previous best bound was O(n 3/2 ).