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1-Planar Graphs

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Beyond Planar Graphs

Abstract

Topological graph theory discusses, in most cases, graphs embedded in the plane (or other surfaces). For example, such plane graphs are sometimes regarded as the simplest town maps. Now, we consider a town having some pedestrian bridges, which cannot be realized by a plane graph. Its underlying graph can actually be regarded as a 1-plane graph. The notion of 1-plane and 1-planar graphs was first introduced by Ringel in connection with the problem of simultaneous coloring of the vertices and faces of plane graphs. In particular, in contrast to planarity testing, testing 1-planarity of a given graph is an NP-complete problem. Even though 1-planar graphs have been widely studied recently, we still know relatively little about them. In this chapter, we begin with formally defining 1-plane and 1-planar graphs and mainly focus on “maximal”, “maximum,” and “optimal” 1-planar graphs, which are relatively easy to treat. This chapter reviews some basic properties of these graphs.

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Acknowledgements

This work was supported by JSPS KAKENHI Grant Number 16K05250.

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Authors and Affiliations

  1. Department of Mathematics, Niigata University, 8050 Ikarashi 2-no-cho, Nishi-ku, Niigata, 950-2181, Japan

    Yusuke Suzuki

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  1. Yusuke Suzuki
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Correspondence to Yusuke Suzuki .

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Editors and Affiliations

  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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Suzuki, Y. (2020). 1-Planar Graphs. In: Hong, SH., Tokuyama, T. (eds) Beyond Planar Graphs. Springer, Singapore. https://doi.org/10.1007/978-981-15-6533-5_4

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  • DOI: https://doi.org/10.1007/978-981-15-6533-5_4

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  • Publisher Name: Springer, Singapore

  • Print ISBN: 978-981-15-6532-8

  • Online ISBN: 978-981-15-6533-5

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