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