On edge-sets of bicliques in graphs

Authors:
Marina Groshaus;Pavol Hell;Juraj Stacho
Affiliations:
Universidad de Buenos Aires, Facultad de Ciencias Exactas y Naturales, DepartamentodeComputación, Buenos Aires, Argentina;Simon Fraser University, School of Computing Science, 8888 University Drive, Burnaby, B.C., Canada V5A 1S6;Wilfrid Laurier University, Department of Physics & Computer Science, 75University Ave W, Waterloo, ON N2L 3C5, Canada
Venue:
Discrete Applied Mathematics
Year:
2012

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Abstract

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.