On Independent Sets and Bicliques in Graphs

Authors:
Serge Gaspers;Dieter Kratsch;Mathieu Liedloff
Affiliations:
Vienna University of Technology, Institute of Information Systems, Favoritenstraße 9-11, 1040, Vienna, Austria;Université Paul Verlaine-Metz, LITA, 57045, Metz Cedex 01, France;Université d’Orléans, LIFO, 45067, Orléans Cedex 2, France
Venue:
Algorithmica
Year:
2012

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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(3 n/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(1.3642 n ), 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.