# Queue Layouts, Tree-Width, and Three-Dimensional Graph Drawing

## Abstract

A *three-dimensional (straight-line grid) drawing* of a graph represents the vertices by points in Z3 and the edges by non-crossing line segments. This research is motivated by the following open problem due to Felsner, Liotta, and Wismath [Graph Drawing ’01, *Lecture Notes in Comput. Sci.*, 2002]: *does every n-vertex planar graph have a threedimensional drawing with O(n) volume?* We prove that this question is almost equivalent to an existing one-dimensional graph layout problem. A *queue layout* consists of a linear order σ of the vertices of a graph, and a partition of the edges into queues, such that no two edges in the same queue are nested with respect to σ. The minimum number of queues in a queue layout of a graph is its *queue-number*. Let G be an n-vertex member of a proper minor-closed family of graphs (such as a planar graph). We prove that G has a O(1) × O(1) × O(n) drawing if and only if G has O(1) queue-number. Thus the above question is almost equivalent to an open problem of Heath, Leighton, and Rosenberg [*SIAM J. Discrete Math.*, 1992], who ask whether every planar graph has O(1) queue-number? We also present partial solutions to an open problem of Ganley and Heath [*Discrete Appl. Math.*, 2001], who ask whether graphs of bounded tree-width have bounded queue-number? We prove that graphs with bounded path-width, or both bounded tree-width and bounded maximum degree, have bounded queue-number. As a corollary we obtain three-dimensional drawings with optimal O(n) volume, for series-parallel graphs, and graphs with both bounded tree-width and bounded maximum degree.

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