A drawing of a graph G is a mapping which assigns to each vertex a point of the plane and to each edge a simple continuous arc connecting the corresponding two points. The crossing number of G is the minimum number of crossing points in any drawing of G. We define two new parameters, as follows. The pairwise crossing number (resp. the odd-crossing number) of G is the minimum number of pairs of edges that cross (resp. cross an odd number of times) over all drawings of G. We prove that the largest of these numbers (the crossing number) cannot exceed twice the square of the smallest (the odd-crossing number). Our proof is based on the following generalization of an old result of Hanani, which is of independent interest. Let G be a graph and let E0 be a subset of its edges such that there is a drawing of G, in which every edge belonging to E0 crosses any other edge an even number of times. Then G can be redrawn so that the elements of E0 are not involved in any crossing. Finally, we show that the determination of each of these parameters is an NP-hard problem and it is NP-complete in the case of the crossing number and the odd-crossing number. 1

Given a finite, undirected, simple graph G, we are concerned with operations on G that transform it into a planar graph. We give a survey of results about such operations and related graph parameters. While there are many algorithmic results about planarization through edge deletion, the results about vertex splitting, thickness, and crossing number are mostly of a structural nature. We also include a brief section on vertex deletion. We do not consider parallel algorithms, nor do we deal with on-line algorithms.

Scheinerman and Wilf [SW94] assert that "an important open problem in the study of graph embeddings is to determine the rectilinear crossing number of the complete graph Kn ." A rectilinear embedding or drawing of Kn is an arrangement of n vertices in the plane, every pair of which is connected by an edge that is a line segment. We assume that no three vertices are collinear. The rectilinear crossing number of Kn is the fewest number of edge crossings attainable over all planar rectilinear embeddings of Kn . For each n we construct a rectilinear embedding of Kn that has the fewest number of edge crossings and the best asymptotics known to date. Moreover, we give some alternative infinite families of embeddings of Kn with good asymptotics. Finally, we mention some old and new open problems.

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.

Scheinerman and Wilf [SW94] assert that "an important open problem in the study of graph embeddings is to determine the rectilinear crossing number of the complete graph Kn ." A rectilinear embedding or drawing of Kn is an arrangement of n vertices in the plane, every pair of which is connected by an edge that is a line segment. We assume that no three vertices are collinear. The rectilinear crossing number of Kn is the fewest number of edge crossings attainable over all planar rectilinear embeddings of Kn . For each n we construct a rectilinear embedding of Kn that has the fewest number of edge crossings and the best asymptotics known to date. Moreover, we give some alternative infinite families of embeddings of Kn with good asymptotics. Finally, we mention some old and new open problems. 1 Introduction and History Given an arbitrary graph G, determining an embedding of G in the plane that produces the fewest number of edge crossings is NP-Complete [GJ83]. The same can be said for a...

A convex drawing of an n-vertex graph G = (V, E) is a drawing in which the vertices are placed on the corners of a convex n-gon in the plane and each edge is drawn using one straight line segment. We derive a general lower bound on the number of crossings in any convex drawings of G, using isoperimetric properties of G. The result implies that convex drawings for many graphs, including the planar 2-dimensional grid on n vertices have at least \Omega(n log n) crossings. Moreover, for any given arbitrary drawing of G with c crossings in the plane, we construct a convex drawing with at most O((c + v ) log n) crossings, where dv is the degree of v.

Graphs are often used to model complex systems and to visualize relationships, and this often involves drawing a graph in the plane. For this, a variety of algorithms and mathematical tools have been used with varying success. We demonstrate why it is often more natural and more meaningful to view higher dimensional representations of graphs. We present some of the theory and problems associated with constructing such representations, and we briefly describe some visualization tools which are now available for experimental research in this area. Figure 1: Picture of a graph 1 Introduction For many people the right picture is the key to understanding. The various ways of visualizing a graph provide different insights, and often hidden relationships are revealed. Also, the representation of a graph enfluences how well it can be used. We focus on several mathematical problems associated with drawing graphs. The problems and methods of solution are as diverse as the objectives includin...

22.88> n. This was proved by Beineke and Ringeisen [refs as before] for m 4. At the recent meeting in Braunschweig, Carsten Thomassen announced that he and Bruce Richter have proved a result from which (C 5 C 5 ) = 15 follows, but they do not have a proof for C 5 C n . The t-linear crossing number t (G) is the minimum number of crossings in a planar drawing of G such that each edge of G is represented by a curve consisting of at most t line segments; 1 (G) is what traditionally was called the rectilinear crossing number. Dan Bienstock (Columbia Univ.) and Nate Dean (Bellcore) have many results about these parameters. In [4], they proved that 1 (G) = (G) if<