2k-Vertex Kernels for Cluster Deletion and Strong Triadic Closure

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.