  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.
