On Independent Sets and Bicliques in Graphs

Serge Gaspers (1) , Dieter Kratsch (2) , Mathieu Liedloff (3)

1 Institute of Information Systems
2 LITA - Laboratoire d'Informatique Théorique et Appliquée
3 LIFO - Laboratoire d'Informatique Fondamentale d'Orléans

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, 2000] and show that the maximum number of maximal bicliques in a graph on $n$ vertices is $\Theta(3^{n/3})$. 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.

Domaines: Algorithme et structure de données [cs.DS]

Serge Gaspers, Dieter Kratsch, Mathieu Liedloff. On Independent Sets and Bicliques in Graphs. Algorithmica, 2012, 62 (3-4), pp.637-658. ⟨10.1007/s00453-010-9474-1⟩. ⟨hal-00536312⟩