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Simultaneous Embedding

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Beyond Planar Graphs

Abstract

Given two planar graphs \(G_1\) and \(G_2\) that share some vertices and edges, a simultaneous embedding with fixed edges (Sefe) is a pair of planar topological drawings \(\varGamma _i\) of \(G_i\), for \(i=1,2\), that coincide on the shared graph \(G_1 \cap G_2\). Despite much progress in the last years, the complexity of the corresponding decision problem is still open. This chapter surveys the developments in this area from the last decade. We first describe the recently discovered relations between the Sefe problem (which asks to decide whether a given pair of graphs admits a Sefe) and several other graph drawing problems, which show that Sefe is one of the most general problems in the context of planarity. Afterward, we survey algorithmic approaches to the Sefe problem, give an overview of recent results, and discuss their limitations. We close with a brief discussion of some recent variations of the simultaneous embedding problem.

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Notes

  1. 1.

    Note that both the drawing one seeks and the problem of deciding whether given input graphs admit such a drawing are called Sefe. This is a somewhat unfortunate double-meaning. On the other hand, the meaning is typically clear, and we follow this convention from the literature.

  2. 2.

    Pach and Wenger [42] show that any planar graph with a fixed combinatorial embedding can be drawn with fixed vertex positions and a linear number of bends per edge. Fixing the positions of all vertices arbitrarily at distinct points in the plane and applying the result by Pach and Wenger independently for both graphs yields the desired drawing.

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Authors and Affiliations

  1. Faculty of Computer Science and Mathematics, University of Passau, Passau, Germany

    Ignaz Rutter

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  1. Ignaz Rutter
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Correspondence to Ignaz Rutter .

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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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Rutter, I. (2020). Simultaneous Embedding. In: Hong, SH., Tokuyama, T. (eds) Beyond Planar Graphs. Springer, Singapore. https://doi.org/10.1007/978-981-15-6533-5_13

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  • DOI: https://doi.org/10.1007/978-981-15-6533-5_13

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