String Graphs with Precise Number of Intersections

A string graph is an intersection graph of curves in the plane. A k-string graph is a graph with a string representation in which every pair of curves intersects in at most k points. We introduce the class of (= k)-string graphs as a further restriction of k-string graphs by requiring that every two curves intersect in either zero or precisely k points. We study the hierarchy of these graphs, showing that for any k >= 1, (= k)-string graphs are a subclass of (= k+ 2)-string graphs as well as of (= 4k)-string graphs; however, there are no other inclusions between the classes of (= k)-string and (= l)-string graphs apart from those that are implied by the above rules. In particular, the classes of (= k)-string graphs and (= k + 1)-string graphs are incomparable by inclusion for any k, and the class of (= 2)-string graphs is not contained in the class of (= 2l + 1)-string graphs for any l.