In the Cluster Deletion problem the goal is to remove the minimum number of edges of a given graph, such that every connected component of the resulting graph constitutes a clique. It is known that the decision version of Cluster Deletion is NP-complete on (\(P_5\)-free) chordal graphs, whereas Cluster Deletion is solved in polynomial time on split graphs. However, the existence of a polynomial-time algorithm of Cluster Deletion on interval graphs, a proper subclass of chordal graphs, remained a well-known open problem. Our main contribution is that we settle this problem in the affirmative, by providing a polynomial-time algorithm for Cluster Deletion on interval graphs. Moreover, despite the simple formulation of a polynomial-time algorithm on split graphs, we show that Cluster Deletion remains NP-complete on a natural and slight generalization of split graphs that constitutes a proper subclass of \(P_5\)-free chordal graphs. Although the later result arises from the already-known reduction for \(P_5\)-free chordal graphs, we give an alternative proof showing an interesting connection between edge-weighted and vertex-weighted variations of the problem. To complement our results, we provide faster and simpler polynomial-time algorithms for Cluster Deletion on subclasses of such a generalization of split graphs.

在“聚类删除”问题中，目标是删除给定图的最小边数，以使结果图的每个连接的组件都组成一个集团。据了解，决定版本集群删除是NP完全（\（P_5 \） -免费）弦图，而集群删除是在分割图多项式时间内解决。但是，区间图上的聚类删除的多项式时间算法（弦图的适当子类）的存在仍然是一个众所周知的开放问题。我们的主要贡献是，我们通过为簇删除提供多项式时间算法来肯定地解决了这个问题。在间隔图上。此外，尽管在拆分图上简单地采用了多项式时间算法，但我们显示，聚类删除在拆分图的自然和轻微泛化上仍然是NP完全的，它构成了\（P_5 \） -无弦图的适当子类。尽管后面的结果来自无已知\（P_5 \）弦图的减少，但我们给出了另一种证明，表明问题的边缘加权和顶点加权变化之间存在有趣的联系。为了补充我们的结果，我们针对此类分裂图的泛化为子类提供了更快，更简单的多项式时间算法，用于聚类删除。