The Complexity of Cluster Vertex Splitting and Company

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.