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On Forbidden Induced Subgraphs for Unit Disk Graphs

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Abstract

A unit disk graph is the intersection graph of disks of equal radii in the plane. The class of unit disk graphs is hereditary, and therefore admits a characterization in terms of minimal forbidden induced subgraphs. In spite of quite active study of unit disk graphs very little is known about minimal forbidden induced subgraphs for this class. We found only finitely many minimal non-unit disk graphs in the literature. In this paper we study in a systematic way forbidden induced subgraphs for the class of unit disk graphs. We develop several structural and geometrical tools, and use them to reveal infinitely many new minimal non-unit disk graphs. Further we use these results to investigate structure of co-bipartite unit disk graphs. In particular, we give structural characterization of those co-bipartite unit disk graphs whose edges between parts form a \(C_4\)-free bipartite graph, and show that bipartite complements of these graphs are also unit disk graphs. Our results lead us to propose a conjecture that the class of co-bipartite unit disk graphs is closed under bipartite complementation.

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Notes

  1. All subgraphs in this paper are induced and further we sometimes omit the word ‘induced’.

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Acknowledgements

The authors are grateful to the anonymous reviewers for their careful reading of the paper and many helpful suggestions and comments. Zamaraev acknowledges support from EPSRC, Grant EP/L020408/1; and from Russian Foundation for Basic Research, Grant 14-01-00515-a.

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Authors and Affiliations

  1. School of Science and Technology, Nottingham Trent University, Nottingham, NG11 8NS, UK

    Aistis Atminas

  2. Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK

    Viktor Zamaraev

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  1. Aistis Atminas
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  2. Viktor Zamaraev
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Correspondence to Viktor Zamaraev.

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Atminas, A., Zamaraev, V. On Forbidden Induced Subgraphs for Unit Disk Graphs. Discrete Comput Geom 60, 57–97 (2018). https://doi.org/10.1007/s00454-018-9968-1

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  • DOI: https://doi.org/10.1007/s00454-018-9968-1

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