Layouts of Graph Subdivisions

Dujmović, Vida and Wood, David R. (2004) Layouts of Graph Subdivisions. In: Graph Drawing 12th International Symposium, GD 2004, September 29-October 2, 2004 , pp. 133-143(Official URL: http://dx.doi.org/10.1007/978-3-540-31843-9_15).

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Abstract

A k-stack layout (respectively, k-queue layout) 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 0(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. Research supported by NSERC and COMBSTRU.

Item Type: Conference Paper 10.1007/978-3-540-31843-9_15 G Algorithms and Complexity > G.420 CrossingsZ Theory > Z.500 Representations http://gdea.informatik.uni-koeln.de/id/eprint/580