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

Abstract

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 $\lim_{k \rightarrow \infty}|V(H^k(G))|=\infty$ (resp. $\lim_{k \rightarrow \infty}H^k(G)=H^m(G)$ for some $m$ or $H^k(G)=H^{k+s}(G)$ for some $k$ and $s \geq 2$). 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 \textit{burgeon graphs} and we propose some general conjectures on the subject.