Vertex removal in biclique graphs

Abstract

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 $H^{\prime }$ such that $G-\{q\}=KB(H^{\prime })$? 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 $H^{\prime }$.