几乎所有的图在biclique算子下都是发散的
Almost every graph is divergent under the biclique operator
摘要：
图$G$的一个双团是图$G$的一个极大导出完全二部子图。用$KB(G)$表示的$G$的二分图是$G$的所有二分图的交图。双团图可以被认为是图之间的一个算子。用$KB^{k}(G)$表示$G$的迭代双团图，它是通过将双团算子$k$连续多次应用于$G$而得到的图。相关的问题是当$k$增长到无穷大时，判定输入图在biclique算子下是收敛、发散还是周期的。最近，所有可能的行为进行了表征，并给出了一个$O(n^4)$算法，以确定在biclique算子下的任何图的行为。在这项工作中，我们证明了biclique图的新的结构结果。特别地，我们证明了每一个至少有$13$顶点的无假孪生图是发散的。这些结果导致一个线性时间算法来解决同样的问题。
英文摘要：
A biclique of a graph $G$ is a maximal induced complete bipartite subgraph of $G$. The biclique graph of $G$ denoted by $KB(G)$, is the intersection graph of all the bicliques of $G$. The biclique graph can be thought as an operator between graphs. The iterated biclique graph of $G$ denoted by $KB^{k}(G)$, is the graph obtained by applying the biclique operator $k$ successive times to $G$. The associated problem is deciding whether an input graph converges, diverges or is periodic under the biclique operator when $k$ grows to infinity. All possible behaviors were characterized recently and an $O(n^4)$ algorithm for deciding the behavior of any graph under the biclique operator was also given. In this work we prove new structural results of biclique graphs. In particular, we prove that every false-twin-free graph with at least $13$ vertices is divergent. These results lead to a linear time algorithm to solve the same problem.
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