Biclique Graphs of $K_3$-free Graphs and Bipartite Graphs

TL;DR

This paper characterizes the biclique graphs of $K_3$-free graphs and bipartite graphs, providing new tools and properties to understand their structure, including a characterization as square graphs of a new graph class.

Contribution

It introduces a characterization of biclique graphs of $K_3$-free graphs as square graphs of a new graph class and characterizes biclique graphs of bipartite graphs as squares of IIC-comparability graphs.

Findings

Biclique graph of $K_3$-free graphs is the square of $KB_m(G)$.

Biclique graphs of bipartite graphs are squares of IIC-comparability graphs.

Generalizes properties about induced ${P_3}$s to stars in biclique graphs.

Abstract

A biclique of a graph is a maximal complete bipartite subgraph. The biclique graph of a graph $G$, $KB(G)$, defined as the intersection graph of the bicliques of $G$, was introduced and characterized in 2010. However, this characterization does not lead to polynomial time recognition algorithms. The time complexity of its recognition problem remains open. There are some works on this problem when restricted to some classes. In this work we give a characterization of the biclique graph of a $K_3$-free graph $G$. We prove that $KB(G)$ is the square graph of a particular graph which we call Mutually Included Biclique Graph of $G$ ($KB_m(G)$). Although it does not lead to a polynomial time recognition algorithm, it gives a new tool to prove properties of biclique graphs (restricted to $K_3$-free graphs) using known properties of square graphs. For instance we generalize a property about induced ${P_3}'$s in biclique graphs to a property about stars and proved a conjecture posted by Groshaus and Montero, when restricted to $K_3$-free graphs. Also we characterize the class of biclique graphs of bipartite graphs. We prove that $KB($bipartite$) = ($IIC-comparability$)^2$, where IIC-comparability is a subclass of comparability graphs that we call Interval Intersection Closed Comparability.