A biclique is a maximal set of vertices in a graph that induces a complete bipartite subgraph. The biclique graph of a graph G is the intersection graph of all bicliques of G and we denote such graph by KB( G ). In this work we introduce the concept of biclique parts of G and the inclusion graph of biclique parts of G, denoted by BP( G ). We show that the class of BP( K 3 –free) graphs is the same as a subclass of comparability graphs which we introduce as skew-IIC comparability graphs, from which we derive a characterization of KB( K 3 –free) graphs. We also present a proper subclass of K 3 –free graphs such that its class of biclique graphs is the same as the class of biclique graphs of all K 3 –free graphs. Furthermore, it is proved that the problem of computing a preimage of a KB m ( K 3 –free) graph can be reduced to a variation of the graph sandwich problem.

A biclique is a maximal set of vertices in a graph that induces a complete bipartite graph. The biclique graph KB(G) of a graph G is the intersection graph of all bicliques in G. In this work, we introduce the concept of differentiating edges and non-edges between pairs of intersecting bicliques in a graph and the corresponding variants of the biclique graph: the edge differentiated (KBedif) and the non-edge differentiated (KBndif) biclique graphs. Two bicliques are mutually included if they can be partitioned respectively into (X1, Y1) and (X2, Y2) such that X1 c X2 and Y2 c Y1. We show that all pairs of mutually included bicliques are non-edge differentiated, but they are not edge differentiated. We show that every pair of intersecting bicliques are differentiated by either edge or non-edge. Finally, we prove that graphs are free of edge differentiated bicliques if and only if they are (K3, C5)-free and that graphs are free of non-edge differentiated bicliques if and only if they are (P4, paw)-free.