Cluster deletion and strong triadic closure are two important NP-complete problems that have received significant attention due to their applications in various areas, including social networks and data analysis. Although cluster deletion and strong triadic closure are closely linked by induced paths on three vertices, there are subtle differences between them. In some cases, the solutions of strong triadic closure and cluster deletion are quite different. In this paper, we study the parameterized algorithms for these two problems. More specifically, we focus on the kernels of these two problems. Instead of separating the critical clique and its neighbors for analysis, we consider them as a whole, which allows us to more effectively bound the number of related vertices. In addition, in analyzing the kernel of strong triadic closure, we introduce the concept of edge-disjoint induced path on three vertices, which enables us to obtain the lower bound of weak edge number in a more concise way. Our analysis demonstrates that cluster deletion and strong triadic closure both admit 2k-vertex kernels. These results represent improvements over previously best-known kernels for both problems. Furthermore, our analysis provides additional insights into the relationship between cluster deletion and strong triadic closure.

聚类删除和强三元闭包是两个重要的 NP 完全问题，由于它们在社交网络和数据分析等各个领域的应用而受到广泛关注。尽管簇删除和强三元闭包通过三个顶点上的诱导路径紧密相连，但它们之间存在细微的差异。在某些情况下，强三元闭包和簇删除的解决方案有很大不同。在本文中，我们研究了这两个问题的参数化算法。更具体地说，我们关注这两个问题的核心。我们没有将关键集团及其邻居分开进行分析，而是将它们视为一个整体，这使我们能够更有效地限制相关顶点的数量。此外，在分析强三元闭包的核心时，我们引入了三个顶点上边不相交诱导路径的概念，这使得我们能够以更简洁的方式获得弱边数的下界。我们的分析表明，簇删除和强三元闭包都允许 2 k顶点内核。这些结果代表了对这两个问题的先前最知名内核的改进。此外，我们的分析为簇删除和强三元闭包之间的关系提供了更多见解。