Vertex removal in biclique graphs

TL;DR

This paper investigates the conditions under which vertices can be removed from biclique graphs while preserving their structure, showing that generally it's not possible but identifying specific cases where it is.

Contribution

The paper proves that vertex removal generally does not preserve biclique graph structure, but identifies and characterizes cases where it is possible, specifically when the vertex degree is two.

Findings

Vertex removal usually does not result in a biclique graph.

If a vertex has degree two, its removal yields a biclique graph.

Provides a method to obtain the corresponding graph H' after removal.

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 $H'$ 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 $H'$.