String graphs with precise number of intersections
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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.
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