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Recognizing Optimal 1-Planar Graphs in Linear Time

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Abstract

A graph with n vertices is 1-planar if it can be drawn in the plane such that each edge is crossed at most once, and is optimal if it has the maximum of \(4n-8\) edges. We show that optimal 1-planar graphs can be recognized in linear time. Our algorithm implements a graph reduction system with two rules, which can be used to reduce every optimal 1-planar graph to an irreducible extended wheel graph. The graph reduction system is non-deterministic, constraint, and non-confluent.

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Acknowledgments

I would like to thank Christian Bachmaier and Josef Reislhuber for many inspiring discussions and their support and the reviewers for their valuable suggestions.

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Authors and Affiliations

  1. University of Passau, 94030, Passau, Germany

    Franz J. Brandenburg

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  1. Franz J. Brandenburg
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Correspondence to Franz J. Brandenburg.

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Supported by the Deutsche Forschungsgemeinschaft (DFG), Grant Br835/18-1.

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Brandenburg, F.J. Recognizing Optimal 1-Planar Graphs in Linear Time. Algorithmica 80, 1–28 (2018). https://doi.org/10.1007/s00453-016-0226-8

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