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 $\ell$.

翻译：弦图是平面上曲线的交叉图。若一个图存在弦表示，使得每对曲线至多相交于$k$个点，则称其为$k$-弦图。我们引入$(=k)$-弦图类作为$k$-弦图的进一步限制，要求每两条曲线要么相交于零个点，要么恰好相交于$k$个点。我们研究了这些图的层次结构，证明了对于任意$k\geq 1$，$(=k)$-弦图既是$(=k+2)$-弦图的子类，也是$(=4k)$-弦图的子类；然而，除了上述规则所蕴含的包含关系外，$(=k)$-弦图类与$(=\ell)$-弦图类之间不存在其他包含关系。特别地，对于任意$k$，$(=k)$-弦图类与$(=k+1)$-弦图类在包含关系下不可比较，且对于任意$\ell$，$(=2)$-弦图类不包含于$(=2\ell+1)$-弦图类。