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Outer 1-Planar Graphs

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A Correction to this article was published on 26 September 2021

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Abstract

A graph is outer 1-planar (o1p) if it can be drawn in the plane such that all vertices are in the outer face and each edge is crossed at most once. o1p graphs generalize outerplanar graphs, which can be recognized in linear time, and specialize 1-planar graphs, whose recognition is \({NP}\)-hard. We explore o1p graphs. Our first main result is a linear-time algorithm that takes a graph as input and returns a positive or a negative witness for o1p. If a graph \(G\) is o1p, then the algorithm computes an embedding and can augment \(G\) to a maximal o1p graph. Otherwise, \(G\) includes one of six minors, which is detected by the recognition algorithm. Secondly, we establish structural properties of o1p graphs. o1p graphs are planar and are subgraphs of planar graphs with a Hamiltonian cycle. They are neither closed under edge contraction nor under subdivision. Several important graph parameters, such as treewidth, colorability, stack number, and queue number, increase by one from outerplanar to o1p graphs. Every o1p graph of size \(n\) has at most \(\frac{5}{2} n - 4\) edges and there are maximal o1p graphs with \(\frac{11}{5} n - \frac{18}{5}\) edges, and these bounds are tight. Finally, every o1p graph has a straight-line grid drawing in \(\fancyscript{O}(n^2)\) area with all vertices in the outer face, a planar visibility representation in \(\fancyscript{O}(n \log n)\) area, and a 3D straight-line drawing in linear volume, and these drawings can be constructed in linear time.

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Notes

  1. A short version of the algorithm appeared in [5].

  2. In the rooted version of SPQR-trees the expansion graph \({{\mathrm{\mathrm {expg}}}}(e)\) corresponds to the pertinent graph of \({{\mathrm{\mathrm {refn}}}}(e)\).

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Acknowledgments

We thank the anonymous referees for their careful reading and useful comments and suggestions.

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Authors and Affiliations

  1. University of Passau, 94030, Passau, Germany

    Christopher Auer, Christian Bachmaier, Franz J. Brandenburg, Andreas Gleißner, Kathrin Hanauer, Daniel Neuwirth & Josef Reislhuber

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  1. Christopher Auer
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  2. Christian Bachmaier
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  3. Franz J. Brandenburg
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  4. Andreas Gleißner
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  5. Kathrin Hanauer
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  6. Daniel Neuwirth
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  7. Josef Reislhuber
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Corresponding author

Correspondence to Christian Bachmaier.

Additional information

This work was supported in part by the Deutsche Forschungsgemeinschaft (DFG) Grant Br835/18-1.

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Auer, C., Bachmaier, C., Brandenburg, F.J. et al. Outer 1-Planar Graphs. Algorithmica 74, 1293–1320 (2016). https://doi.org/10.1007/s00453-015-0002-1

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