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Algorithms for 1-Planar Graphs

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Beyond Planar Graphs

Abstract

A 1-planar graph is a graph that can be embedded in the plane with at most one crossing per edge. It is known that testing 1-planarity of a graph is NP-complete. This chapter reviews the algorithmic results on 1-planar graphs. We first review a linear time algorithm for testing maximal 1-planarity of a graph if a rotation system (i.e., the circular ordering of edges for each vertex) is given. A graph is maximal 1-planar if the addition of an edge destroys 1-planarity. Next, we sketch a linear time algorithm for testing outer-1-planarity. A graph is outer-1-planar if it has an embedding in which every vertex is on the outer face and each edge has at most one crossing. The 1-plane graphs have two forbidden subgraphs to admit a straight-line drawing. We review a linear time algorithm for constructing a straight-line drawing of 1-plane graphs. Finally, we conclude with reviews on recent related results.

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Acknowledgements

This work is supported by ARC (Australian Research Council) Discovery Project grant.

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  1. University of Sydney, Sydney, Australia

    Seok-Hee Hong

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  1. Seok-Hee Hong
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Correspondence to Seok-Hee Hong .

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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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Hong, SH. (2020). Algorithms for 1-Planar Graphs. In: Hong, SH., Tokuyama, T. (eds) Beyond Planar Graphs. Springer, Singapore. https://doi.org/10.1007/978-981-15-6533-5_5

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  • DOI: https://doi.org/10.1007/978-981-15-6533-5_5

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