We give a survey of techniques for deriving lower bounds and algorithms for constructing upper bounds for several variations of the crossing number problem. Our aim is to emphasize the more general results or those results which have an algorithmic flavor, including the recent results of the authors. We also show applications of crossing numbers to other areas of discrete mathematics, like discrete geometry.

We also prove by similar methods that a graph G with crossing number k = cr(G) ? Cpssqd(G) m log2 n has a nonplanar subgraph on at most O\Gamma \Delta nm log2 nk \Delta vertices, where m is the number of edges, \Delta is the maximum degree in G, and C is a suitable sufficiently large constant.

We propose the recursive grid layout scheme for deriving efficient layouts of a variety of hierarchical networks and computing upper bounds on the VLSI area of general hierarchical networks. In particular, we construct optimal VLSI layouts for butterfly networks, generalized hypercubes, and star graphs that have areas within a factor of 1 + o#1# from their lower bounds. We also derive efficient layouts for a number of other important networks, such as cubeconnected cycles (CCC) and hypernets, which are the best results reported for these networks thus far.

Abstract. According to Euler’s formula, every planar graph with n vertices has at most O(n) edges. How much can we relax the condition of planarity without violating the conclusion? After surveying some classical and recent results of this kind, we prove that every graph of n vertices, which can be drawn in the plane without three pairwise crossing edges, has at most O(n) edges. For straight-line drawings, this statement has been established by Agarwal et al., using a more complicated argument, but for the general case previously no bound better than O(n 3/2) was known. 1

The following problem was raised by M. Watanabe. Let P be a self-intersecting closed polygon with n vertices in general position. How manys steps does it take to untangle P , i.e., to turn it into a simple polygon, if in each step we can arbitrarily relocate one of its vertices. It is shown that in some cases one has to move all but at most O((n log n) 2=3 ) vertices. On the other hand, every polygon P can be untangled in at most n p n) steps. Some related questions are also considered. 1

In this paper, we present efficient VLSI layouts of several hypercubic networks. We show that an N-node hypercube and an N-node cube-connected cycles (CCC) graph can be laid out in 4N²/9 + o(N²) and 4N²/(9log²_2 N) + o(N²/log² N) areas, respectively, both of which are optimal within a factor of 1.7 + o(1). We introduce the multilayer grid model, and present efficient layouts of hypercubes that use more than 2 layers of wires. We derive efficient layouts for butterfly networks, generalized hypercubes, hierarchical swapped networks, and indirect swapped networks, that are optimal within a factor of 1+o(1). We also present efficient layouts for folded hypercubes, reduced hypercubes, recursive hierarchical swapped networks, and enhanced-cubes, which are the best results reported for these networks thus far.

Abstract not found

We survey known results and propose open problems on the biplanar crossing number. We study biplanar crossing numbers of specific families of graphs, in particular, of complete bipartite graphs. We find a few particular exact values and give general lower and upper bounds for the biplanar crossing number. We find the exact biplanar crossing number of K 5;q for every q.

Interconnection networks play a vital role in parallel computing architectures. We investigate topological properties of some networks proposed for parallel computation, based on their underlying graph models. The vertices of the graph correspond to processors and the edges represent communication links between processors. Parameters such as crossing number and thickness strongly affect the area required to lay out the corresponding circuit on a VLSI chip. In particular, we give upper bounds for the skewness, crossing number, and thickness of several networks including the mesh of trees, reduced mesh of trees, 2-dimensional torus, butterfly, wrapped butterfly, and Benes graph.

Introduction Several interconnection networks of parallel computers based on permutations have appeared recently, e.g. bubblesort and star graph, alternating group graphs ..., see a survey paper of Lakshmivarahan et al. [8]. Leighton [10] introduced the n\Gammadimensional complete transposition graph CT n : It consists of n! vertices, each of which corresponds to a permutation on n numbers. Two vertices are adjacent iff their corresponding permutations differ by a single transposition. Leighton suggested to study a computational power of a parallel computer with the complete transposition graph interconnection network. Especially, he asked what the bisection width of the complete transposition graph is (problem (R) 3.356). The bisection width is the size of the smallest edge cut of a graph which divides it into two equal parts. This graph invariant is a fundamental concept in the theory of