A \textit{biclique} is a maximal induced complete bipartite subgraph of $G$. The \textit{biclique graph} of a graph $G$, denoted by $KB(G)$, is the intersection graph of the family of all bicliques of $G$. In this work we study some structural properties of biclique graphs which are necessary conditions for a graph to be a biclique graph. In particular, we prove that for biclique graphs that are neither a $K_3$ nor a \textit{diamond}, the number of vertices of degree $2$ is less than half the number of vertices in the graph. Also, we present forbidden structures. For this, we introduce a natural definition of the distance between bicliques in a graph. We give a formula that relates the distance between bicliques in a graph $G$ and the distance between their respective vertices in $KB(G)$. Using these results, we can prove not only this new necessary condition involving the degree, but also that some graphs are not biclique graphs. For example, we show that the \textit{crown} is the smallest graph that is not a biclique graph although the known necessary condition for biclique graphs holds, answering an open problem about biclique graphs. Finally, we present some interesting related conjectures and open problems.
翻译:\ textit{ biclique} 是一个最大导出完整的双叶分包 $G$。 由 $KB (G) 表示的图形 $G$ 的\ textit{ biclique 图形} 是所有双球 $G$ 的家族的交叉图。 在此工作中, 我们研究双球图的一些结构属性, 这些属性对于图形成为双球图来说是必要的条件。 特别是, 我们证明, 对于既不是 K_ 3$ 或\ textit{ diamon} 的双球图来说, 度 $2 的顶点数量小于图中顶点数的一半。 另外, 我们展示了被禁止的结构。 我们用一个自然定义来定义双球图之间的距离。 我们给出了一个公式, 将一个图形中的 $G$( G ) 和 各自在 $KB (G) 的张点之间的距离联系起来。 使用这些结果, 我们可以证明, 度的顶点不仅仅是这个新的需要条件, 与图表中的双曲线 。 虽然这个颜色 图表是一定的颜色 。