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arXiv · 2308.15590

String graphs with precise number of intersections

Petr Chmel·Vít Jelínek
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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 $\ell$.

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Petr Chmel, Vít Jelínek. 2023-08-29. String graphs with precise number of intersections. https://arxiv.org/abs/2308.15590

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