On the Relation of Strong Triadic Closure and Cluster Deletion

We study the parameterized and classical complexity of two related problems on undirected graphs G=(V,E). In Strong Triadic Closure we aim to label the edges in E as strong and weak such that at most~k edges are weak and G contains no induced P3 with two strong edges. In Cluster Deletion, we aim to destroy all induced P3s by a minimum number of edge deletions. We first show that Strong Triadic Closure admits a 4k-vertex kernel. Then, we study parameterization by ℓ:=|E|-k and show that both problems are fixed-parameter tractable and unlikely to admit a polynomial kernel with respect to ℓ. Finally, we give a dichotomy of the classical complexity of both problems on H-free graphs for all H of order four.