Clustering a graph when the clusters can overlap can be seen from three different angles: We may look for cliques that cover the edges of the graph, we may look to add or delete few edges to uncover the cluster structure, or we may split vertices to separate the clusters from each other. Splitting a vertex $v$ means to remove it and to add two new copies of $v$ and to make each previous neighbor of $v$ adjacent with at least one of the copies. In this work, we study the underlying computational problems regarding the three angles to overlapping clusterings, in particular when the overlap is small. We show that the above-mentioned covering problem, which also has been independently studied in different contexts,is NP-complete. Based on a previous so-called critical-clique lemma, we leverage our hardness result to show that Cluster Editing with Vertex Splitting is also NP-complete, resolving an open question by Abu-Khzam et al. [ISCO 2018]. We notice, however, that the proof of the critical-clique lemma is flawed and we give a counterexample. Our hardness result also holds under a version of the critical-clique lemma to which we currently do not have a counterexample. On the positive side, we show that Cluster Vertex Splitting admits a vertex-linear problem kernel with respect to the number of splits.

翻译：当图聚类允许簇重叠时，可从三个不同角度审视：寻找覆盖图边的团簇、通过增删少量边揭示簇结构、或通过分裂顶点将簇彼此分离。顶点$v$的分裂操作是指移除该顶点并添加两个新副本，使$v$的每个原邻接顶点至少与其中一个副本相邻。本工作研究这三类重叠聚类视角下的底层计算问题，特别关注重叠度较小的情况。我们证明上述覆盖问题（该问题此前已在不同语境中被独立研究）是NP完全的。基于先前所谓的临界团引理，我们利用该困难性结果证明带顶点分裂的聚类编辑问题同样为NP完全，由此解决了Abu-Khzam等人[ISCO 2018]提出的公开问题。然而我们注意到临界团引理的证明存在缺陷，并给出反例。我们的困难性结果在目前尚无反例的临界团引理版本下依然成立。正面结果是，我们证明集群顶点分裂问题关于分裂次数存在顶点线性问题核。