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Clique-Width: Harnessing the Power of Atoms

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Graph-Theoretic Concepts in Computer Science (WG 2020)

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Abstract

Many NP-complete graph problems are polynomial-time solvable on graph classes of bounded clique-width. Several of these problems are polynomial-time solvable on a hereditary graph class \(\mathcal{G}\) if they are so on the atoms (graphs with no clique cut-set) of \(\mathcal{G}\). Hence, we initiate a systematic study into boundedness of clique-width of atoms of hereditary graph classes. A graph G is H-free if H is not an induced subgraph of G, and it is \((H_1,H_2)\)-free if it is both \(H_1\)-free and \(H_2\)-free. A class of H-free graphs has bounded clique-width if and only if its atoms have this property. This is no longer true for \((H_1,H_2)\)-free graphs, as evidenced by one known example. We prove the existence of another such pair \((H_1,H_2)\) and classify the boundedness of clique-width on \((H_1,H_2)\)-free atoms for all but 18 cases.

The research in this paper received support from the Leverhulme Trust (RPG-2016-258). Masařík and Novotná were supported by Charles University student grants (SVV-2017-260452 and GAUK 1277018) and GAČR project (17-09142S). The last author was supported by Polish National Science Centre grant no. 2018/31/D/ST6/00062.

A preprint of the full version of this paper is available from arXiv  [29].

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Notes

  1. 1.

    See Sect. 2 for a definition of clique-width and other terminology used in Sect. 1.

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Authors and Affiliations

  1. Department of Computer Science, Durham University, Durham, UK

    Konrad K. Dabrowski & Daniël Paulusma

  2. Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic

    Tomáš Masařík & Jana Novotná

  3. Institute of Informatics, University of Warsaw, Warsaw, Poland

    Tomáš Masařík, Jana Novotná & Paweł Rzążewski

  4. Department of Mathematics, Simon Fraser University, Burnaby, Canada

    Tomáš Masařík

  5. Faculty of Mathematics and Information Science, Warsaw University of Technology, Warsaw, Poland

    Paweł Rzążewski

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  1. Konrad K. Dabrowski
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  2. Tomáš Masařík
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  3. Jana Novotná
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  4. Daniël Paulusma
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  5. Paweł Rzążewski
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Correspondence to Konrad K. Dabrowski .

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  1. University of Leeds, Leeds, UK

    Isolde Adler

  2. University of Leeds, Leeds, UK

    Haiko Müller

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Dabrowski, K.K., Masařík, T., Novotná, J., Paulusma, D., Rzążewski, P. (2020). Clique-Width: Harnessing the Power of Atoms. In: Adler, I., Müller, H. (eds) Graph-Theoretic Concepts in Computer Science. WG 2020. Lecture Notes in Computer Science(), vol 12301. Springer, Cham. https://doi.org/10.1007/978-3-030-60440-0_10

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