簇编辑问题的新的时间解释
A New Temporal Interpretation of Cluster Editing
AI总结：
本文提出时间图视角下的簇编辑问题，通过禁止配置特征设计了FPT算法，解决了静态图中无法用有限顶点集合描述的特征问题。
摘要：
NP完全的图问题簇编辑旨在通过最少的边修改将静态图转换为不相交的 clique union。我们引入了这一问题在时间图中的自然解释，其边集随时间变化。即使在限制为时间图的底层图是路径的情况下，该问题仍然是 NP 完全的，但我们得到了两种多项式时间算法。在静态情况下，已知一个图是不相交 clique 的 union 当且仅当它不包含诱导的 P_3 复制；我们证明在时间设置中不存在最多四个顶点的集合的一般特征，但获得了一个涉及最多五个顶点的禁止配置的完整特征。该特征导致了一个参数化于允许修改数量和时间图寿命的 FPT 算法。
英文摘要：
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.
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