In an edge modification problem, we are asked to modify at most $k$ edges to 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: \begin{itemize} \item a $2 k$-vertex kernel for the cluster edge deletion problem, \item a $3 k^2$-vertex kernel for the trivially perfect completion problem, \item a $5 k^{1.5}$-vertex kernel for the split completion problem and the split edge deletion problem, and \item a $5 k^{1.5}$-vertex kernel for the pseudo-split completion problem and the pseudo-split edge deletion problem. \end{itemize} 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.
翻译:在边缘修改问题中, 我们被要求修改最多为 $k$ 的图形边缘, 以使图表满足一定的属性。 根据允许的操作, 我们存在完成问题和边缘删除问题。 大量的努力都致力于理解这些问题的内分解复杂性。 我们重新研究了一些研究周密的边缘修改问题, 并为它们开发更好的内核 :\ begin{ filtiziz}\ 项一个 2 k$ 的顶端内核, 用于分组边缘删除问题 。\ end{ hitsizize} 此外, 我们用于分解完成和假顶端的顶端内核, 用于小的完美完成问题 3 k%2$ 的顶端内核, \ i 项目5 k ⁇ 1.5} 美元 用于分解完成问题 和 分裂边缘删除问题 。 我们的顶层内核也包含一个与关闭有关的 $ koffex iel 问题 。