The number of convergent graphs under the biclique operator with no twin vertices is finite

The biclique graph of G, KB(G), is the intersection graph of the bicliques of G. Given a graph G, the iterated biclique graph of G, KBk(G), is the graph defined iteratively as follows: KBk+1(G)=KB(KBk(G)). Say that a graph G diverges (resp. converges) under the operator KB whenever limk→∞V(KBk(G))=∞ (resp. limk→∞KBk(G)=KBm(G) for some m). Each of these behaviours were recently characterized. These characterizations lead to a O(n4) time algorithm for deciding the divergence or convergence of a graph. In this work we prove that any graph with at least 7 bicliques diverges under the biclique operator. Furthermore, we prove that graphs with no twin vertices that are not divergent have at most 12 vertices, which leads to a linear time algorithm to decide if a graph converges or diverges under the biclique operator.