Editing a graph to obtain a disjoint union of s-clubs is one of the models for correlation clustering, which seeks a partition of the vertex set of a graph so that elements of each resulting set are close enough according to some given criterion. For example, in the case of editing into s-clubs, the criterion is proximity since any pair of vertices (in an s-club) are within a distance of s from each other. In this work we consider the vertex splitting operation, which allows a vertex to belong to more than one cluster. This operation was studied as one of the parameters associated with the Cluster Editing problem. We study the complexity and parameterized complexity of the s-Club Cluster Edge Deletion with Vertex Splitting and s-Club Cluster Vertex Splitting problems. Both problems are shown to be NP-Complete and APX-hard. On the positive side, we show that both problems are Fixed-Parameter Tractable with respect to the number of allowed editing operations and that s-Club Cluster Vertex Splitting is solvable in polynomial-time on the class of forests.

翻译：将图编辑为若干s-club的不交并是相关性聚类的一种建模方法，该方法旨在对图的顶点集进行划分，使得每个结果集中的元素根据给定准则具有足够紧密的关联。例如，在编辑为s-club的情形中，其准则是邻近性，因为（s-club中的）任意顶点对之间的距离不超过s。本研究考虑顶点分割操作，该操作允许一个顶点同时属于多个聚类。该操作曾作为聚类编辑问题的关联参数之一被研究。我们探讨了s-Club聚类边删除（含顶点分割）与s-Club聚类顶点分割两个问题的计算复杂性及参数化复杂性。研究证明这两个问题均属于NP完全问题且具有APX难度。在积极方面，我们证明了这两个问题对于允许的编辑操作数量具有固定参数可解性，并且s-Club聚类顶点分割问题在森林图类上存在多项式时间解法。