On Independent Sets and Bicliques in Graphs

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 vert...

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Bibliographic Details
Published in:Algorithmica Vol. 62; no. 3-4; pp. 637 - 658
Main Authors: Gaspers, Serge, Kratsch, Dieter, Liedloff, Mathieu
Format: Journal Article
Language:English
Published: New York Springer-Verlag 01.04.2012
Springer
Springer Verlag
Subjects:
Algorithm Analysis and Problem Complexity
Algorithms
Applied sciences
Computer Science
Computer science; control theory; systems
Computer Systems Organization and Communication Networks
Data Structures and Algorithms
Data Structures and Information Theory
Exact sciences and technology
Information retrieval. Graph
Mathematics of Computing
Theoretical computing
Theory of Computation
Exact exponential time algorithm
Maximal independent sets
Maximal bicliques
Counting algorithms
Combinatorial bound
Branching
Independent graph
Polynomial time
Upper bound
Bicyclic graph
Independent set
Maximal graph
Time complexity
Combinatorial analysis
Counting
ISSN:0178-4617, 1432-0541
Online Access:Get full text
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Summary: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.
ISSN:0178-4617
1432-0541
DOI:10.1007/s00453-010-9474-1