On the edge-biclique graph and the iterated edge-biclique operator

08/19/2019

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by   Sylvain Legay, et al.

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A biclique of a graph G is a maximal induced complete bipartite subgraph of G. The edge-biclique graph of G, KB_e(G), is the edge-intersection graph of the bicliques of G. A graph G diverges (resp. converges or is periodic) under an operator H whenever lim_k →∞|V(H^k(G))|=∞ (resp. lim_k →∞H^k(G)=H^m(G) for some m or H^k(G)=H^k+s(G) for some k and s ≥ 2). The iterated edge-biclique graph of G, KB_e^k(G), is the graph obtained by applying the edge-biclique operator k successive times to G. In this paper, we first study the connectivity relation between G and KB_e(G). Next, we study the iterated edge-biclique operator KB_e. In particular, we give sufficient conditions for a graph to be convergent or divergent under the operator KB_e, we characterize the behavior of burgeon graphs and we propose some general conjectures on the subject.