Plane Drawings of Queue and Deque Graphs

Abstract

Abstract. In stack and queue layouts the vertices of a graph are linearly ordered from left to right, where each edge corresponds to an item and the left and right end vertex of each edge represents the addition and removal of the item to the used data structure. A graph admitting a stack or queue layout is a stack or queue graph, respectively.Typical stack and queue layouts are rainbows and twists visualizing the LIFO and FIFO principles, respectively. However, in such visualizations, twists cause many crossings, which make the drawings incomprehensible. We introduce linear cylindric layouts as a visualization technique for queue and deque (double-ended queue) graphs. It provides new insights into the characteristics of these fundamental data structures and extends to the visualization of mixed layouts with stacks and queues. Our main result states that a graph is a deque graph if and only if it has a plane linear cylindric drawing.

Characterizing Planarity by the Splittable Deque

“…1(b) and (d), which crosses edge e 1 . Edge e 1 is inserted at the head at vertex 1 and e at the tail at vertex 2, i. e., e 1 e . At vertex 4, the deque has to be split such that e 1 ∈ c 10 and e ∈ c 5 .…”

“…Each edge is inserted and removed exactly once. Further, in the loop from lines 5-12 each forward edge is processed at most twice during the algorithm and, by using the DFS numbers, it is possible to decide in time O (1) in which subtree the end vertex of a forward edge lies. Also, each vertex is inserted at most twice to the stack for each edge.…”

“…In [1,2], we have studied double-ended queue (deque) layouts: A deque has two ends, a head and a tail, and items can be inserted and removed at both sides. It can emulate two stacks and additionally allows for queue edges, i. e., edges inserted and removed at opposite sides.…”

Classification of Planar Upward Embedding

“…C divides C r into a left-and a right-hand region. To avoid a crossing between the edges (1, 4) and (2, 5), they must lie in different regions, e. g., (1,4) to the right and (2,5) to the left of C. Now consider the region R enclosed by the edges (1, 2), (2,5), (4,5), (1,4), which contains vertex 3. The curve of edge (3, 1) must start within R and, due to the homogeneous field, must reach vertex 1 from below.…”

“…Another subclass of RUP are the graphs with a queue layout, see [1]. The input-output behavior of a queue is represented by a graph such that the behavior is legal if and only if the graph has a RUP embedding with all vertices placed on a horizontal line.…”