Bicliques In Graph Research Articles

Efficient Mining of Maximal Bicliques in Graph by Pruning Search Space

In this paper, we present a new algorithm for mining or enumerating maximal biclique (MB) subgraphs in an undirected general graph. Our algorithm achieves improved theoretical efficiency in time over the best algorithms. For an undirected graph with n vertices, m edges and k maximal bicliques, our algorithm requires O(kn2) time, which is the state of the art performance. Our main idea is based on a strategy of pruning search space extensively. This strategy is made possible by the approach of storing maximal bicliques immediately after detection and allowing them to be looked up during runtime to make pruning decisions. The space complexity of our algorithm is O(kn) because of the space used for storing the MBs. However, a lot of space is saved by using a compact way of storing MBs, which is an advantage of our method. Experiments show that our algorithm outperforms other state of the art methods.

On edge-sets of bicliques in graphs

A biclique is a maximal induced complete bipartite subgraph of a graph. We investigate the intersection structure of edge-sets of bicliques in a graph. Specifically, we study the associated edge-biclique hypergraph whose hyperedges are precisely the edge-sets of all bicliques. We characterize graphs whose edge-biclique hypergraph is conformal (i.e., it is the clique hypergraph of its 2-section) by means of a single forbidden induced obstruction, the triangular prism. Using this result, we characterize graphs whose edge-biclique hypergraph is Helly and provide a polynomial time recognition algorithm. We further study a hereditary version of this property and show that it also admits polynomial time recognition, and, in fact, is characterized by a finite set of forbidden induced subgraphs. We conclude by describing some interesting properties of the 2-section graph of the edge-biclique hypergraph.

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 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.