## 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 Stacks, Queues and Tracks: Layouts of Graph Subdivisions 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. If your browser does not display the abstract correctly (because of the different mathematical symbols) you can look it up in the PostScript or PDF files. 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 For a corresponding BibTeX entry, please consider our BibTeX-file. dm070111.ps.gz (468 K) dm070111.ps (1767 K) dm070111.pdf (549 K)