On the Relation of Strong Triadic Closure and Cluster Deletion

We study the parameterized and classical complexity of two problems that are concerned with induced paths on three vertices, called  $$P_3$$ P 3 s, in undirected graphs  $$G=(V,E)$$ 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  $$P_3$$ P 3 with two strong edges. In Cluster Deletion we aim to destroy all induced  $$P_3$$ P 3 s by a minimum number of edge deletions. We first show that Strong Triadic Closure admits a 4 k -vertex kernel. Then, we study parameterization by  $$\ell :=|E|-k$$ ℓ : = | E | - k and show that both problems are fixed-parameter tractable and unlikely to admit a polynomial kernel with respect to  $$\ell $$ ℓ . Finally, we give a dichotomy of the classical complexity of both problems on  H -free graphs for all  H of order at most four.