Vertex removal in biclique graphs

A \textit{biclique} is a maximal induced complete bipartite subgraph. The \textit{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 $q$ of $G$, such that $G - \{q\}$ is a biclique graph? And if possible, can we obtain a graph $H'$ such that $G - \{q\} = KB(H')$? We show that the general question has a "no" for answer. However, we prove that if $G$ has a vertex $q$ such that $d(q) = 2$, then $G-\{q\}$ is a biclique graph and we show how to obtain $H'$.

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