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On Independent Sets and Bicliques in Graphs

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Abstract

Bicliques of graphs have been studied extensively, partially motivated by the large number of applications. In this paper we improve Prisner’s upper bound on the number of maximal bicliques (Combinatorica, 20, 109–117, 2000) and show that the maximum number of maximal bicliques in a graph on n vertices is Θ(3n/3). Our major contribution is an exact exponential-time algorithm. This branching algorithm computes the number of distinct maximal independent sets in a graph in time O(1.3642n), where n is the number of vertices of the input graph. We use this counting algorithm and previously known algorithms for various other problems related to independent sets to derive algorithms for problems related to bicliques via polynomial-time reductions.

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Authors and Affiliations

  1. Institute of Information Systems, Vienna University of Technology, Favoritenstraße 9-11, 1040, Vienna, Austria

    Serge Gaspers

  2. LITA, Université Paul Verlaine-Metz, 57045, Metz Cedex 01, France

    Dieter Kratsch

  3. LIFO, Université d’Orléans, 45067, Orléans Cedex 2, France

    Mathieu Liedloff

Authors
  1. Serge Gaspers
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  2. Dieter Kratsch
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  3. Mathieu Liedloff
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Corresponding author

Correspondence to Serge Gaspers.

Additional information

A large part of the research was done while Serge Gaspers was visiting the University of Metz. A preliminary version of this paper appeared in the proceedings of WG 2008 [18]. Serge Gaspers acknowledges partial support of NFR and of Conicyt Chile via the project Basal-CMM.

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Gaspers, S., Kratsch, D. & Liedloff, M. On Independent Sets and Bicliques in Graphs. Algorithmica 62, 637–658 (2012). https://doi.org/10.1007/s00453-010-9474-1

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  • DOI: https://doi.org/10.1007/s00453-010-9474-1

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