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International Conference on Algorithms and Complexity

CIAC 2013: Algorithms and Complexity pp 276–287Cite as

Cliques and Clubs

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7878))

Abstract

Clubs are generalizations of cliques. For a positive integer s, an s-club in a graph G is a set of vertices that induces a subgraph of G of diameter at most s. The importance and fame of cliques are evident, whereas clubs provide more realistic models for practical applications. Computing an s-club of maximum cardinality is an NP-hard problem for every fixed s ≥ 1, and this problem has attracted significant attention recently. We present new positive results for the problem on large and important graph classes. In particular we show that for input G and s, a maximum s-club in G can be computed in polynomial time when G is a chordal bipartite or a strongly chordal or a distance hereditary graph. On a superclass of these graphs, weakly chordal graphs, we obtain a polynomial-time algorithm when s is an odd integer, which is best possible as the problem is NP-hard on this class for even values of s. We complement these results by proving the NP-hardness of the problem for every fixed s on 4-chordal graphs, a superclass of weakly chordal graphs. Finally, if G is an AT-free graph, we prove that the problem can be solved in polynomial time when s ≥ 2, which gives an interesting contrast to the fact that the problem is NP-hard for s = 1 on this graph class.

This work is supported by the European Research Council and the Research Council of Norway.

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Author information

Authors and Affiliations

  1. Department of Informatics, University of Bergen, Norway

    Petr A. Golovach, Pinar Heggernes & Arash Rafiey

  2. LITA, Université de Lorraine - Metz, France

    Dieter Kratsch

Authors
  1. Petr A. Golovach
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  2. Pinar Heggernes
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  3. Dieter Kratsch
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  4. Arash Rafiey
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Editor information

Editors and Affiliations

  1. School of Engineering, Department of Computer Engineering and Informatics, University of Patras, 265 00, Patras, Greece

    Paul G. Spirakis

  2. LSI Department, Edif Omega, Universitat Politècnica de Catalunya, Campus Nord, Jordi Girona 1-3, 08034, Barcelona, Spain

    Maria Serna

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Golovach, P.A., Heggernes, P., Kratsch, D., Rafiey, A. (2013). Cliques and Clubs. In: Spirakis, P.G., Serna, M. (eds) Algorithms and Complexity. CIAC 2013. Lecture Notes in Computer Science, vol 7878. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38233-8_23

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  • DOI: https://doi.org/10.1007/978-3-642-38233-8_23

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-38232-1

  • Online ISBN: 978-3-642-38233-8

  • eBook Packages: Computer ScienceComputer Science (R0)

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