Stack and Queue Layouts of Directed Acyclic Graphs : Extended

Stack layouts and queue layouts of undirected graphs are used to model problems in fault tolerant computing, in VLSI design, and in managing the ow of data in a parallel processing system. In certain applications, such as managing the ow of data in a parallel processing system, it is more realistic to use layouts of directed acyclic graphs (dags) as a model. A stack layout of a dag consists of a topological ordering of the nodes of the graph along with an assignment of arcs to stacks such that if the nodes are laid out in a line according to and the arcs are all drawn above the line, then no two arcs that are assigned to the same stack cross. A queue layout is deened similarly, except that arcs are assigned to queues with the condition that no two arcs assigned to a queue nest. The stacknumber of a dag is the smallest number of stacks required for its stack layout, while the queuenumber of a dag is the smallest number of queues required for its queue layout. Classes of dags identiied by the structure of their underlying undirected graphs are studied. We determine the stacknumber and queuenumber of classes of dags whose underlying undirected graphs are trees or unicyclic graphs. We give a forbidden subgraph characterization of those dags with queuenumber equal to 1 whose underlying undirected graphs are trees. We discuss classes of planar dags and outerplanar dags that have queuenumber and stacknumber that diier markedly. Finally, we show that the problems of determining whether a dag can be laid out in 7 queues and of determining whether a dag can be laid out in 9 stacks are both NP-complete. A k-stack layout of a directed acyclic graph (a dag) ~ G = (V; ~ E) consists of a topological ordering of the nodes of ~ G and an assignment of each arc of ~ G to one of k stacks s 1 ; s 2 ; : : : ; s k. Each stack s j obeys the last in rst out discipline and operates as follows. The nodes in V are scanned in the order given by. When a node v is encountered, any arcs assigned to s j that have node v as their head must be at the top of the stack and are popped. Any arcs that are assigned to s j …