Covergence and divergence of the iterated biclique graph

Abstract

A biclique of a graph G is a maximal induced complete bipartite subgraph of G. The biclique graph of G, denoted by KB(G), is the intersection graph of the bicliques of G. We say that a graph G diverges (or converges or is periodic) under an operator F whenever limk→∞ |V (F k (G))| = ∞ (limk→∞ F k (G) = F m(G) for some m, or F k (G) = F k+s(G) for some k and s ≥ 2, respectively). Given a graph G, the iterated biclique graph of G, denoted by KBk (G), is the graph obtained by applying the biclique operator k successive times to G. In this article, we study the iterated biclique graph of G. In particular, we classify the different behaviors of KBk (G) when the number of iterations k grows to infinity. That is, we prove that a graph either diverges or converges under the biclique operator. We give a forbidden structure characterization of convergent graphs, which yield a polynomial time algorithm to decide if a given graph diverges or converges. This is in sharp contrast with the situsation for the better known clique operator, where it is not even known if the corresponding problem is decidable.