GD 2004: Graph Drawing pp 133-143

# Layouts of Graph Subdivisions

• Vida Dujmović
• David R. Wood
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3383)

## 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 $$\mathcal{O}({\rm log} n) to \mathcal{O}({\rm log min}\{sn(G), qn(G)\})$$. 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 $$\mathcal{O}({\rm log}{\sf qn}(G))$$, 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 $$\mathcal{O}({\rm log}{\sf qn}(G))$$.

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## Authors and Affiliations

• Vida Dujmović
• 1
• 2
• David R. Wood
• 2
• 3
1. 1.School of Computer ScienceMcGill UniversityMontréalCanada
2. 2.School of Computer ScienceCarleton UniversityOttawaCanada
3. 3.Department of Applied MathematicsCharles UniversityPragueCzech Republic