Crossing Patterns of Segments

Given a system S of simple continuous curves (``strings'') in the plane, we can define a graph GS as follows. Assign a vertex to each curve, and connect two vertices by an edge if and only if the corresponding two curves intersect. GS is called the intersection graph of S. Not every graph is an intersection graph of a system of curves [EET76] (see Fig. 1 for a simple example). This implies that only a very small fraction of all the 2( n 2) labeled graphs on n vertices have this property. For systems of segments, using standard techniques from real algebraic geometry (see [GP86, AKS90, PPW90]), we obtain a fairly good quantitative result.