Algorithmica

, Volume 62, Issue 3–4, pp 637–658 | Cite as

On Independent Sets and Bicliques in Graphs

Article

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.

Keywords

Counting algorithms Combinatorial bound Maximal bicliques Maximal independent sets Exact exponential time algorithm 

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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Serge Gaspers
    • 1
  • Dieter Kratsch
    • 2
  • Mathieu Liedloff
    • 3
  1. 1.Institute of Information SystemsVienna University of TechnologyViennaAustria
  2. 2.LITAUniversité Paul Verlaine-MetzMetz Cedex 01France
  3. 3.LIFOUniversité d’OrléansOrléans Cedex 2France

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