Stacks, queues and tracks: layouts of graph subdivisions

A κ-stack layout (respectively, κ-queue layout) of a graph consists of a total order of the vertices, and a partition of the edges into κ sets of non-crossing (non-nested) edges with respect to the vertex ordering. A κ-track layout of a graph consists of a vertex κ-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 κ such that G has a κ-stack (κ-queue, κ-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 κ-stack, κ-queue, and κ-track subdivisions, for all values of κ. 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.