Layout of Graphs with Bounded Tree-Width

A queue layout of a graph consists of a total order of the vertices, and a partition of the edges into 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 queue-number. A threedimensional (straight-line grid) drawing of a graph represents the vertices by points in Z and the edges by non-crossing line-segments. This paper contributes three main results: (1) It is proved that the minimum volume of a 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)×O(1)×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, which represents the largest known class of graphs with such drawings. The proofs depend upon our results regarding track layouts and tree-partitions of graphs, which may be of independent interest. keywords: queue layout, queue-number, three-dimensional graph drawing, tree-partition, tree-partition-width, tree-width, k-tree, track layout, track-number, star colouring, star chromatic number. ‡School of Computer Science, McGill University, Montréal, Canada. E-mail: [email protected] §School of Computer Science, Carleton University, Ottawa, Canada. E-mail: {morin,davidw}@scs.carleton.ca ∗Results in this paper were presented at the GD ’02 [26] and FSTTCS ’02 [79] conferences. Research supported by NSERC and FCAR.