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String Graphs with Precise Number of Intersections

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Graph Drawing and Network Visualization (GD 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14465))

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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 \(k\ge 1\), \(({=}\,k)\)-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 \(\ell \).

The first author was supported by the Czech Science Foundation Grant No. 19-27871X. The second author was supported by the Czech Science Foundation Grant No. 23-04949X.

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Authors and Affiliations

  1. Computer Science Institute, Charles University, Prague, Czech Republic

    Petr Chmel & Vít Jelínek

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  1. Petr Chmel
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  2. Vít Jelínek
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Correspondence to Petr Chmel .

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Editors and Affiliations

  1. University Ioannina, Ioannina, Greece

    Michael A. Bekos

  2. University of Ioannina, Osnabrück, Germany

    Markus Chimani

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Chmel, P., Jelínek, V. (2023). String Graphs with Precise Number of Intersections. In: Bekos, M.A., Chimani, M. (eds) Graph Drawing and Network Visualization. GD 2023. Lecture Notes in Computer Science, vol 14465. Springer, Cham. https://doi.org/10.1007/978-3-031-49272-3_6

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  • DOI: https://doi.org/10.1007/978-3-031-49272-3_6

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-49271-6

  • Online ISBN: 978-3-031-49272-3

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