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Extending Partial Representations of Circular-Arc Graphs

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Graph-Theoretic Concepts in Computer Science (WG 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13453))

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Abstract

The partial representation extension problem generalizes the recognition problem for classes of graphs defined in terms of geometric representations. We consider this problem for circular-arc graphs, where several arcs are predrawn and we ask whether this partial representation can be completed. We show that this problem is NP-complete for circular-arc graphs, answering a question of Klavík et al. (2014).

We complement this hardness with tractability results of the representation extension problem for various subclasses of circular-arc graphs. We give linear-time algorithms for extending normal proper Helly and proper Helly representations. For normal Helly circular-arc representations we give an \(\mathcal{O}(n^3)\)-time algorithm where n is the number of vertices.

Surprisingly, for Helly representations, the complexity hinges on the seemingly irrelevant detail of whether the predrawn arcs have distinct or non-distinct endpoints: In the former case the algorithm for normal Helly circular-arc representations can be extended, whereas the latter case turns out to be \(\textsf{NP}\)-complete. We also prove that the partial representation extension problem for unit circular-arc graphs is NP-complete.

Funded by the grant 19-17314J of the GA ČR and by grant Ru 1903/3-1 of the German Science Foundation (DFG). Peter Zeman was also supported by the Swiss National Science Foundation project PP00P2-202667.

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Acknowledgement

We thank Bartosz Walczak for inspiring comments, in particular for his hint to extend Theorem 1 to the case of \(\textsf{CAR}\) with distinct endpoints.

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Authors and Affiliations

  1. Department of Applied Mathematics, Charles University, Prague, Czech Republic

    Jiří Fiala

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

    Ignaz Rutter & Peter Stumpf

  3. Institut de mathématiques, Université de Neuchâtel, Neuchâtel, Switzerland

    Peter Zeman

Authors
  1. Jiří Fiala
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  2. Ignaz Rutter
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  3. Peter Stumpf
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  4. Peter Zeman
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Correspondence to Peter Stumpf .

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Editors and Affiliations

  1. University of Ioannina, Ioannina, Greece

    Michael A. Bekos

  2. Universität Tübingen, Tübingen, Germany

    Michael Kaufmann

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Fiala, J., Rutter, I., Stumpf, P., Zeman, P. (2022). Extending Partial Representations of Circular-Arc Graphs. In: Bekos, M.A., Kaufmann, M. (eds) Graph-Theoretic Concepts in Computer Science. WG 2022. Lecture Notes in Computer Science, vol 13453. Springer, Cham. https://doi.org/10.1007/978-3-031-15914-5_17

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  • DOI: https://doi.org/10.1007/978-3-031-15914-5_17

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