Stack and Queue Layouts for Toruses and Extended Hypercubes S. Bettayeb Department of Computer Science University of Houston-Clear Lake Houston, TX 77058, U.S.A Bettayeb@uchl.edu L. Morales Department of Computer Science University of Houston-Clear Lake Houston, TX 77058, U.S.A MoralesL@uchl.edu M. H. Heydari Department of Computer Science James Madison University Harrisonburg, VA 22807, U.S.A heydarmh@jmu.edu I. H. Sudborough Department of Computer Science University of Texas at Dallas Dallas, TX 75080, U.S.A hal@utdallas.edu Abstract Linear layouts play an important role in many applications including networks and VLSI design. Stack and queue layouts are two important types of linear layouts. We consider the stack number, s(G), and queue number, q(G), for multidimensional k-ary hypercubes and toruses. Heath, Leighton, and Rosenberg showed that d-dimensional ternary hypercubes have stack number Ω(N 1/9 ), with N=3 d nodes. Malitz showed that E edges implies stack number O(√ ܧ). For k-ary d-dimensional hypercubes, with N = k d vertices, Malitz’s bound is O(k d/2 ). We improve this to 2 d+1 -3. The 2 d+1 -3 bound holds for arbitrary d-dimensional toruses. The queue number of d-dimensional k-ary hypercubes or toruses is bounded by O(d). Hence, Heath, Leighton, and Rosenberg exhibit an exponential tradeoff between s(G) and q(G) for multidimensional ternary hypercubes. Conversely, they conjectured that, for any G, q(G) is O(s(G)). We present a family {H} of modified multidimensional toruses and conjecture that q(H) is not O(s(H)). 1. Introduction Graph embeddings and linear layouts of graphs play an important role in a wide variety of applications. Stack layouts and queue layouts are two specific types of linear layouts that are useful in the study of VLSI design, routing and graph drawing, parallel processing and matrix computation, and permutation sorting [1,2,3,4,5,8,9,10,11,13,14]. In this paper, we give upper bounds on the stack number and queue number of k-dimensional toruses and hypercubes. A stack layout of a graph, also known as a book embedding, is a linear layout of the vertices of a graph along the spine of a book and an assignment of edges to stacks or pages so that edges assigned to the same stack do not intersect. The minimum number of stacks in which a graph can be embedded is its stack number, denoted by s(G). In the literature, the terms pagenumber and page embedding are sometimes used instead of stack number and stack embedding, respectively. A queue layout of a graph is another type of linear layout of the vertices of a graph. In this case, the edges of the graph are assigned to queues in such a way that no queue contains a pair of nested edges. The minimum number of queues needed to embed a graph is called its queue number, denoted by q(G). We show that for any d-dimensional torus or extended hypercube the stack number is bounded by 2 d+1 -3. Our upper bound is of interest for multidimensional toruses and extended hypercubes in which some, or all, of the dimension sizes are odd integers. Heath, Leighton, and Rosenberg [6] have shown that a d-dimensional ternary hypercube has stack number Ω(3 n/9 ). Their lower bound shows that the stack number, when each dimension has size 3, must be exponential in the number of dimensions. On the other hand, it is known that when the dimensions are all even integers, the stack number grows linearly with the number of dimensions. Thus, the stack number of d-dimensional toruses and extended hypercubes depends strongly on the parity of its dimensions. We show why this is true and describe a layout technique with sequential “corrections” of the order of vertices that mitigates the problem for the case when dimensions are odd. Basically, the technique modifies the standard layout of alternating left-to-right and right-to-left segments with an amortization of “corrections” that allows the reverse order to be realized without paying the penalty of all edges in the first-to-last “wraparound” connections to be simultaneously pair-wise crossing. So, the advantage of amortization is that the total number of stacks is 1 Proceedings of the 43rd Hawaii International Conference on System Sciences - 2010 978-0-7695-3869-3/10 $26.00 © 2010 IEEE