On the Relation of Strong Triadic Closure and Cluster Deletion

Abstract

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