Stack and Queue Layouts of Directed Acyclic Graphs: Part I

Abstract

Stack layouts and queue layouts of undirected graphs have been used to model problems in fault tolerant computing and in parallel process scheduling. However, problems in parallel process scheduling are more accurately modeled by stack and queue layouts of directed acyclic graphs (dags). A stack layout of a dag is similar to a stack layout of an undirected graph, with the additional requirement that the nodes of the dag be in some topological order. A queue layout is de ned in an analogous manner. The stacknumber (queuenumber) of a dag is the smallest number of stacks (queues) required for its stack layout (queue layout). In this paper, bounds are established on the stacknumber and queuenumber of two classes of dags: tree dags and unicyclic dags. In particular, any tree dag can be laid out in 1 stack and in at most 2 queues; and any unicyclic dag can be laid out in at most 2 stacks and in at most 2 queues. Forbidden subgraph characterizations of 1-queue tree dags and 1queue cycle dags are also presented. Part II of this paper presents algorithmic results|in particular, linear time algorithms for recognizing 1-stack dags and 1-queue dags and proof of NP-completeness for the problem of recognizing a 4-queue dag and the problem of recognizing a 9-stack dag.

Plane Drawings of Queue and Deque Graphs

“…When moving this vertical line like a scan line further to the right, its crossings with edges always correspond to the content of the deque. If the vertical line passes a crossing between two edges e and e , e.g., (3,8) and (4,7), then this can be interpreted as swapping the positions of e and e in the deque, which is an invalid deque operation.…”

“…Stack and queue layouts have been studied extensively in the past, e.g., in [1,2,[4][5][6][7][8][9][12][13][14], and are used for 3D drawings of graphs [12,13], in VLSI design [2] and in other application scenarios (see [9] for a short survey). In these layouts the vertices of a graph are linearly ordered from the left to the right.…”

Volume Requirements of 3D Upward Drawings

“…We complete this section by studying the relationship between upward track layout and another well-known graph parameter, namely the upward queuenumber [15,16,17].…”

“…In this section we study compact 3D upward drawings of trees and paths. We recall that Heath et al [17] proved that every tree DAG has an upward 2-queue layout and that every path DAG has an upward 1-queue layout.…”

Laying Out Iterated Line Digraphs Using Queues

“…Queue and stack layouts of a digraph are similarly defined as those of the underlying graph of the digraph. (Note that our definitions for digraphs are different from queue and stack layouts defined in [19,20] in which all arcs must have the same direction with respect to the vertex ordering. )…”