String graphs with precise number of intersections

TL;DR

This paper introduces and studies the hierarchy of $(=k)$-string graphs, a special class of string graphs where each pair of curves intersects in either zero or exactly $k$ points, revealing complex inclusion relations.

Contribution

It defines the class of $(=k)$-string graphs and analyzes their hierarchical relationships, establishing new inclusion and incomparability results among these classes.

Findings

$(=k)$-string graphs are subclasses of $(=k+2)$- and $(=4k)$-string graphs.

No other class inclusions exist between different $(=k)$-string graph classes beyond the established rules.

$(=k)$-string graphs and $(=k+1)$-string graphs are incomparable for any $k$.

Abstract

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 $k\geq 1$, $(=k)$-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 $(=\ell)$-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 $(=2\ell+1)$-string graphs for any $\ell$.