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. Herein, 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 look at the underlying computational problems regarding the three angles to overlapping clusterings, in particular when the overlap is small. We first show that the above-mentioned covering problem, which also has been independently studied in different contexts, is NP-hard. 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-hard, 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 allowed splits.

翻译：当簇可以重叠时，图的聚类可以从三个不同角度理解：寻找覆盖图边的团，通过增删少量边来揭示簇结构，或通过分割顶点来分离各簇。其中，分割顶点$v$指移除该顶点并添加两个新副本，同时使$v$的每个邻点至少与其中一个副本相邻。本研究聚焦于重叠聚类的三个角度相关的计算问题，特别是当重叠较小时的情形。首先证明上述覆盖问题（该问题也在不同背景下被独立研究）是NP难的。基于先前的所谓关键团引理，我们利用该困难性结果证明带顶点分割的簇编辑问题也是NP难的，从而解答了Abu-Khzam等人[ISCO 2018]提出的开放问题。然而，我们发现关键团引理的证明存在缺陷并给出了反例。我们的困难性结果在目前尚无反例的关键团引理版本下仍然成立。在积极方面，我们证明簇顶点分割问题在允许分割次数方面具有顶点线性问题核。