Queue number

In mathematics, the queue number of a graph is a graph invariant defined analogously to stack number (book thickness) using first-in first-out (queue) orderings in place of last-in first-out (stack) orderings.

A queue layout of a given graph is defined by a total ordering of the vertices of the graph together with a partition of the edges into a number of "queues". The set of edges in each queue is required to avoid edges that are properly nested: if ab and cd are two edges in the same queue, then it should not be possible to have a < c < d < b in the vertex ordering. The queue number qn(G) of a graph G is the minimum number of queues in a queue layout.

Equivalently, from a queue layout, one could process the edges in a single queue using a queue data structure, by considering the vertices in their given ordering, and when reaching a vertex, dequeueing all edges for which it is the second endpoint followed by enqueueing all edges for which it is the first endpoint. The nesting condition ensures that, when a vertex is reached, all of the edges for which it is the second endpoint are ready to be dequeued. Another equivalent definition of queue layouts involves embeddings of the given graph onto a cylinder, with the vertices placed on a line in the cylinder and with each edge wrapping once around the cylinder. Edges that are assigned to the same queue are not allowed to cross each other, but crossings are allowed between edges that belong to different queues.

Queue layouts were defined by Heath & Rosenberg (1992), by analogy to previous work on book embeddings of graphs, which can be defined in the same way using stacks in place of queues. As they observed, these layouts are also related to earlier work on sorting permutations using systems of parallel queues, and may be motivated by applications in VLSI design and in communications management for distributed algorithms.

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