Vertex removal in biclique graphs
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A biclique is a maximal bipartite complete induced subgraph. The biclique graph of a graph H, denoted by KB(H), is the intersection graph of the family of all bicliques of H. In this work we address the following question: Given a biclique graph G=KB(H), is it possible to remove a vertex β of G, such that G - {β} is a biclique graph? And if possible, can we obtain a graph H' such that G - {β} = KB(H')? We show that the general question has a “no” for answer. However, we prove that if G has a vertex β such that d(β) = 2, then G-{β} is a biclique graph and we show how to obtain H'.
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