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Biclique graphs and biclique matrices. (English) Zbl 1216.05104
Summary: A biclique of a graph $$G$$ is a maximal induced complete bipartite subgraph of $$G$$. Given a graph $$G$$, the biclique matrix of $$G$$ is a $$\{0,1,-1\}$$ matrix having one row for each biclique and one column for each vertex of $$G$$, and such that a pair of $$1, -1$$ entries in a same row corresponds exactly to adjacent vertices in the corresponding biclique. We describe a characterization of biclique matrices, in similar terms as those employed in Gilmore’s characterization of clique matrices. On the other hand, the biclique graph of a graph is the intersection graph of the bicliques of $$G$$. Using the concept of biclique matrices, we describe a Krausz-type characterization of biclique graphs. Finally, we show that every induced $$P_{3}$$ of a biclique graph must be included in a diamond or in a 3-fan and we also characterize biclique graphs of bipartite graphs.

##### MSC:
 05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.)
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