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

The biclique graph of G, K B ( G ) , is the intersection graph of the bicliques of G. Given a graph G, the iterated biclique graph of G, K B k ( G ) , is the graph defined iteratively as follows: K B k + 1 ( G ) = K B ( K B k ( G ) ) . Say that a graph G diverges (resp. converges) under the operator KB whenever lim k → ∞ V ( K B k ( G ) ) = ∞ (resp. lim k → ∞ K B k ( G ) = K B m ( G ) for some m). Each of these behaviours were recently characterized. These characterizations lead to a O ( n 4 ) 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.