The Speeds of Families of Intersection Graphs

A fundamental question of graph theory lies in counting the number of graphs which satisfy certain properties. In particular, the structure of intersection graphs of planar curves is unknown, and Schaefer, Sedgwick, and Štefankovič have shown that recognizing such graphs is NP-complete. As such, the number of intersection graphs of planar curves poses an interesting problem. Pach and Tóth have previously proven bounds on the number of intersection graphs of string graphs. We investigate the more specific case of the number of intersection graphs on n vertices of systems of segments of certain algebraic curves, including parabolas, conic sections, polynomials, and rational functions. We extend the results of Pach and Solymosi, who obtained upper bounds on the number of intersection graphs of line segments, and Fox, who obtained tight lower bounds. For each system we establish a set of polynomials whose sign patterns give an intersection graph. We use Warren’s Theorem to obtain an upper bound on the number of sign patterns of this set. We then use a constructive approach to calculate matching lower bounds on the number of intersection graphs. In general, the bounds on the intersection graphs of these systems is nn( f+o(1)), where f is the degree of freedom.