# Stacks, Queues and Tracks: Layouts of Graph Subdivisions

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1 School of Computer Science Ottawa 2 Departament de Matemàtica Aplicada II

Abstract : A \emphk-stack layout respectively, \emphk-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 \emphk-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 \emphstack-number respectively, \emphqueue-number, \emphtrack-number of a graph G, denoted by snG qnG, tnG, is the minimum k such that G has a k-stack k-queue, k-track layout.\par 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 Olog n to Olog min\snG,qnG\. This result reduces the question of whether queue-number is bounded by stack-number to whether 3-stack graphs have bounded queue number.\par 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 Olog qnG, is optimal to within a constant factor, for every graph G. \par 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 Olog qnG bends per edge. Finally, we establish a tight relationship between queue layouts and so-called 2-track thickness of bipartite graphs. \par

Keywords : 2-track thickness book-thickness page-number graph layout graph drawing track layout stack layout queue layout book embedding track-number queue-number stack-number three-dimensional graph drawing geometric thickness subdivision

Author: ** Vida Dujmović - David Wood - **

Source: https://hal.archives-ouvertes.fr/