The NP-complete graph problem Cluster Editing seeks to transform a static graph into a disjoint union of cliques by making the fewest possible edits to the edges. We introduce a natural interpretation of this problem in temporal graphs, whose edge sets change over time. This problem is NP-complete even when restricted to temporal graphs whose underlying graph is a path, but we obtain two polynomial-time algorithms for restricted cases. In the static setting, it is well-known that a graph is a disjoint union of cliques if and only if it contains no induced copy of $P_3$; we demonstrate that no general characterisation involving sets of at most four vertices can exist in the temporal setting, but obtain a complete characterisation involving forbidden configurations on at most five vertices. This characterisation gives rise to an FPT algorithm parameterised simultaneously by the permitted number of modifications and the lifetime of the temporal graph.

翻译：NP完全的图问题“聚类编辑”旨在通过尽可能少的边编辑操作，将静态图转换为不相交的团并集。我们引入该问题在时序图（其边集随时间变化）中的自然解释。即使限制于底层图为路径的时序图，该问题仍为NP完全，但我们针对限制情形获得了两个多项式时间算法。在静态设定中，众所周知一个图是不相交的团并集当且仅当它不含$P_3$导出子图；我们证明在时序设定中不可能存在涉及最多四个顶点集合的一般性刻画，但获得了涉及最多五个顶点禁止构型的完整刻画。该刻画导出一个参数化FPT算法，其参数同时为允许修改次数和时序图的生命周期。