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Beyond Planar Graphs pp 109–130Cite as

\(\textit{\textbf{k}}\)-Planar Graphs

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Abstract

A topological graph is called k-planar, for \(k \ge 0\), if each edge has at most k crossings; hence, by definition, 0-planar topological graphs are plane. An abstract graph is called k-planar if it is isomorphic to a k-planar topological graph, i.e., if it can be drawn on the plane with at most k crossings per edge. While planar and 1-planar graphs have been extensively studied in the literature and their structure has been well understood, this is not the case for k-planar graphs, with \(k \ge 2\). These graphs have a more complex structure, which is significantly more difficult to comprehend. As an example, we mention that tight (possibly up to additive constants) bounds on the edge-density of k-planar graphs are only known for small values of k (that is, for \(k\in \{0,1,2,3,4\}\)), even though their existence yields corresponding improvements on the leading constant of the lower bound on the number of crossings of a graph, provided by the well-known Crossing Lemma. In this chapter, we focus on k-planar graphs, with \(k \ge 2\), and review the known combinatorial and algorithmic results from the literature. We also identify several interesting open problems in the field.

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Notes

  1. 1.

    Recall that the dual \(G^*\) of a plane graph G is defined as follows: \(G^*\) has a vertex for each face of G and for every two vertices of \(G^*\) there is an edge connecting them if and only if there corresponding faces of G share an edge.

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Authors and Affiliations

  1. Wilhelm-Schickard-Institut für Informatik, Universität Tübingen, Tübingen, Germany

    Michael A. Bekos

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  1. Michael A. Bekos
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Correspondence to Michael A. Bekos .

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  1. School of Computer Science, University of Sydney, Sydney, NSW, Australia

    Seok-Hee Hong

  2. School of Science and Technology, Kwansei Gakuin University, Sanda, Hyogo, Japan

    Takeshi Tokuyama

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Bekos, M.A. (2020). \(\textit{\textbf{k}}\)-Planar Graphs. In: Hong, SH., Tokuyama, T. (eds) Beyond Planar Graphs. Springer, Singapore. https://doi.org/10.1007/978-981-15-6533-5_7

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  • DOI: https://doi.org/10.1007/978-981-15-6533-5_7

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