On edge-sets of bicliques in graphs

TL;DR

This paper explores the structure of edge-sets of bicliques in graphs, characterizing certain hypergraph properties like conformality and Helly-ness, and providing polynomial algorithms for recognition based on forbidden subgraphs.

Contribution

It introduces a characterization of graphs with conformal and Helly edge-biclique hypergraphs using forbidden induced subgraphs, with polynomial recognition algorithms.

Findings

Characterization of graphs with conformal edge-biclique hypergraphs via the triangular prism

Polynomial-time algorithms for recognizing Helly and hereditary properties

Finite forbidden subgraph characterization for hereditary properties

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.