Discrete Mathematics & Theoretical Computer Science, Vol 7 (2005)

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DMTCS vol 7 no 1 (2005), pp. 155-202

Discrete Mathematics & Theoretical Computer Science


Volume 7 n° 1 (2005), pp. 155-202

author:Vida Dujmović and David R. Wood
title:Stacks, Queues and Tracks: Layouts of Graph Subdivisions
keywords:graph layout, graph drawing, track layout, stack layout, queue layout, book embedding, track-number, queue-number, stack-number, page-number, book-thickness, 2-track thickness, geometric thickness, subdivision, three-dimensional graph drawing

A k-stack layout (respectively, k-queuelayout) of a graph consists of a total order of the vertices, and a partition of the edges into k sets of non-crossing (non-nested) edges with respect to the vertex ordering. A k-track layout of a graph consists of a vertex k-colouring, and a total order of each vertex colour class, such that between each pair of colour classes no two edges cross. The stack-number (respectively, queue-number, track-number) of a graph G, denoted by sn(G) (qn(G), tn(G)), is the minimum k such that G has a k-stack (k-queue, k-track) layout.

This paper studies stack, queue, and track layouts of graph subdivisions. It is known that every graph has a 3-stack subdivision. The best known upper bound on the number of division vertices per edge in a 3-stack subdivision of an n-vertex graph G is improved from O(log n) to O(log min{sn(G),qn(G)}). This result reduces the question of whether queue-number is bounded by stack-number to whether 3-stack graphs have bounded queue number.

It is proved that every graph has a 2-queue subdivision, a 4-track subdivision, and a mixed 1-stack 1-queue subdivision. All these values are optimal for every non-planar graph. In addition, we characterise those graphs with k-stack, k-queue, and k-track subdivisions, for all values of k. The number of division vertices per edge in the case of 2-queue and 4-track subdivisions, namely O(log qn(G)), is optimal to within a constant factor, for every graph G.

Applications to 3D polyline grid drawings are presented. For example, it is proved that every graph G has a 3D polyline grid drawing with the vertices on a rectangular prism, and with O(log qn(G)) bends per edge. Finally, we establish a tight relationship between queue layouts and so-called 2-track thickness of bipartite graphs.

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reference: Vida Dujmović and David R. Wood (2005), Stacks, Queues and Tracks: Layouts of Graph Subdivisions, Discrete Mathematics and Theoretical Computer Science 7, pp. 155-202
bibtex:For a corresponding BibTeX entry, please consider our BibTeX-file.
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