Improved Kernels for Edge Modification Problems

Abstract

In an edge modification problem, we are asked to modify at most k edges of a given graph to make the graph satisfy a certain property. Depending on the operations allowed, we have the completion problems and the edge deletion problems. A great amount of efforts have been devoted to understanding the kernelization complexity of these problems. We revisit several well-studied edge modification problems, and develop improved kernels for them: a 2 k -vertex kernel for the cluster edge deletion problem, a 3 k 2 -vertex kernel for the trivially perfect completion problem, a 5 k 1.5 -vertex kernel for the split completion problem and the split edge deletion problem, and a 5 k 1.5 -vertex kernel for the pseudo-split completion problem and the pseudo-split edge deletion problem. Moreover, our kernels for split completion and pseudo-split completion have only O ( k 2.5 ) edges. Our results also include a 2 k -vertex kernel for the strong triadic closure problem, which is related to cluster edge deletion.